74e48a5...master · laop

git.iko.soy/laop

37 files changed+1573−3662

since splitsnapshots

unifiedsplit

.ghc.environment.x86_64-linux-8.6.5−17 .github/workflows/ci.yml+43 .gitignore+54−2 .travis.yml+47 CHANGELOG.md+17 ChangeLog.md−3 LICENSE+17−26 README.md+35−27 Setup.hs−2 app/Main.hs−7 app/Spreadsheets.hs−61 appveyor.yml+34 app/Benchmarks.hs → benchmark/Main.hs+16−18 hackage.sh−10 laop-0.1.0.0-docs.tar.gz laop.cabal+89 package.yaml−57 report-lazy.html−1169 report-lazy.pdf report-strict.html−1169 report-strict.pdf src/LAoP/Dist.hs+2−2 src/LAoP/Dist/Internal.hs+46−45 src/LAoP/Matrix/Internal.hs+371−400 src/LAoP/Matrix/Nat.hs+139−124 src/LAoP/Matrix/Type.hs+231−175 src/LAoP/Relation.hs+8−9 src/LAoP/Relation/Internal.hs+245−234 src/LAoP/Utils.hs+4−1 src/LAoP/Utils/Internal.hs+28−2 stack-8.8.3.yaml+1 stack.yaml+1−66 stack.yaml.lock+4−4 test/Examples/Quantum.hs+31 app/Examples.hs → test/Examples/Readme.hs+56−31 test/Examples/Spreadsheets.hs+47 test/Spec.hs+7−1

.ghc.environment.x86_64-linux-8.6.5deleted−17

@@ -1,17 +0,0 @@

1−-- This is a GHC environment file written by cabal. This means you can

2−-- run ghc or ghci and get the environment of the project as a whole.

3−-- But you still need to use cabal repl $target to get the environment

4−-- of specific components (libs, exes, tests etc) because each one can

5−-- have its own source dirs, cpp flags etc.

6−--

7−clear-package-db

8−global-package-db

9−package-db /home/bolt/.cabal/store/ghc-8.6.5/package.db

10−package-db packagedb/ghc-8.6.5

11−package-id laop-0.1.0.0-inplace

12−package-id base-4.12.0.0

13−package-id ghc-prim-0.5.3

14−package-id rts

15−package-id integer-gmp-1.0.2.0

16−package-id deepseq-1.4.4.0

17−package-id array-0.5.3.0

.github/workflows/ci.ymlnew file+43

@@ -0,0 +1,43 @@

1+name: CI

3+# Trigger the workflow on push or pull request, but only for the master branch

4+on:

5+ pull_request:

6+ push:

7+ branches: [master]

9+jobs:

10+ build:

11+ name: ghc ${{ matrix.ghc }}

12+ runs-on: ubuntu-16.04

13+ strategy:

14+ matrix:

15+ cabal: ["3.0"]

16+ ghc:

17+ - "8.6.5"

18+ - "8.8.3"

19+

20+ steps:

21+ - uses: actions/checkout@v2

22+ if: github.event.action == 'opened' || github.event.action == 'synchronize' || github.event.ref == 'refs/heads/master'

23+

24+ - uses: actions/setup-haskell@v1

25+ name: Setup Haskell

26+ with:

27+ ghc-version: ${{ matrix.ghc }}

28+ cabal-version: ${{ matrix.cabal }}

29+

30+ - uses: actions/cache@v1

31+ name: Cache ~/.cabal/store

32+ with:

33+ path: ~/.cabal/store

34+ key: ${{ runner.os }}-${{ matrix.ghc }}-cabal

35+

36+ - name: Build

37+ run: |

38+ cabal v2-update

39+ cabal v2-build --enable-tests --enable-benchmarks

40+

41+ - name: Test

42+ run: |

43+ cabal v2-test --enable-tests

.gitignore+54−2

@@ -1,3 +1,55 @@

1+### Haskell

2+dist

3+dist-*

4+cabal-dev

5+*.o

6+*.hi

7+*.chi

8+*.chs.h

9+*.dyn_o

10+*.dyn_hi

11+*.prof

12+*.aux

13+*.hp

14+*.eventlog

15+.virtualenv

16+.hsenv

17+.hpc

18+.cabal-sandbox/

19+cabal.sandbox.config

20+cabal.config

21+cabal.project.local

22+.ghc.environment.*

23+.HTF/

24+# Stack

125.stack-work/

2−laop.cabal

3−*~

26+stack.yaml.lock

27+

28+### IDE/support

29+# Vim

30+[._]*.s[a-v][a-z]

31+[._]*.sw[a-p]

32+[._]s[a-v][a-z]

33+[._]sw[a-p]

34+*~

35+tags

36+

37+# IntellijIDEA

38+.idea/

39+.ideaHaskellLib/

40+*.iml

41+

42+# Atom

43+.haskell-ghc-mod.json

44+

45+# VS

46+.vscode/

47+

48+# Emacs

49+*#

50+.dir-locals.el

51+TAGS

52+

53+# other

54+.DS_Store

55+hackage.sh

.travis.ymlnew file+47

@@ -0,0 +1,47 @@

1+sudo: true

2+language: haskell

4+git:

5+ depth: 5

7+cabal: "3.0"

9+cache:

10+ directories:

11+ - "$HOME/.cabal/store"

12+ - "$HOME/.stack"

13+ - "$TRAVIS_BUILD_DIR/.stack-work"

14+

15+matrix:

16+ include:

17+ - ghc: 8.6.5

18+ - ghc: 8.8.3

19+

20+ - ghc: 8.8.3

21+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack-8.8.3.yaml"

22+

23+ - ghc: 8.6.5

24+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack.yaml"

25+

26+

27+install:

28+ - |

29+ if [ -z "$STACK_YAML" ]; then

30+ cabal v2-update

31+ cabal v2-build --enable-tests --enable-benchmarks

32+ else

33+ curl -sSL https://get.haskellstack.org/ | sh

34+ stack --version

35+ stack build --system-ghc --test --bench --no-run-tests --no-run-benchmarks

36+ fi

37+

38+script:

39+ - |

40+ if [ -z "$STACK_YAML" ]; then

41+ cabal v2-test --enable-tests

42+ else

43+ stack test --system-ghc

44+ fi

45+

46+notifications:

47+ email: false

CHANGELOG.mdnew file+17

@@ -0,0 +1,17 @@

1+# Changelog

3+`laop` uses [PVP Versioning][1].

4+The changelog is available [on GitHub][2].

6+## 0.1.1.1

8+* Bump base version to work with cabal

10+## 0.1.1.0

11+

12+* Package is more organized and now has tests and benchmarks separated

13+* Repository is more organized and clean

14+* Added CI

15+

16+[1]: https://pvp.haskell.org

17+[2]: https://github.com/bolt12/laop2/releases

ChangeLog.mddeleted−3

@@ -1,3 +0,0 @@

1−# Changelog for laop

2−

3−## Unreleased changes

LICENSE+17−26

@@ -1,30 +1,21 @@

1+MIT License

5−Redistribution and use in source and binary forms, with or without

6−modification, are permitted provided that the following conditions are met:

5+Permission is hereby granted, free of charge, to any person obtaining a copy

6+of this software and associated documentation files (the "Software"), to deal

7+in the Software without restriction, including without limitation the rights

8+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

9+copies of the Software, and to permit persons to whom the Software is

10+furnished to do so, subject to the following conditions:

711

8− * Redistributions of source code must retain the above copyright

9− notice, this list of conditions and the following disclaimer.

12+The above copyright notice and this permission notice shall be included in all

13+copies or substantial portions of the Software.

1014

11− * Redistributions in binary form must reproduce the above

12− copyright notice, this list of conditions and the following

13− disclaimer in the documentation and/or other materials provided

14− with the distribution.

15−

16− * Neither the name of Armando Santos nor the names of other

17− contributors may be used to endorse or promote products derived

18− from this software without specific prior written permission.

19−

20−THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS

21−"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

22−LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR

23−A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT

24−OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,

25−SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT

26−LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,

27−DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY

28−THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT

29−(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE

30−OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

15+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

16+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

17+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE

18+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

19+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

20+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE

21+SOFTWARE.

README.md+35−27

@@ -1,5 +1,11 @@

11# laop - Linear Algebra of Programming library

3+[![GitHub CI](https://github.com/bolt12/laop/workflows/CI/badge.svg)](https://github.com/bolt12/laop/actions)

4+[![Build status](https://img.shields.io/travis/bolt12/laop.svg?logo=travis)](https://travis-ci.org/bolt12/laop)

5+[![Hackage](https://img.shields.io/hackage/v/laop.svg?logo=haskell)](https://hackage.haskell.org/package/laop)

6+[![Stackage Lts](http://stackage.org/package/laop/badge/lts)](http://stackage.org/lts/package/laop)

7+[![Stackage Nightly](http://stackage.org/package/laop/badge/nightly)](http://stackage.org/nightly/package/laop)

39The LAoP discipline generalises relations and functions treating them as

410Boolean matrices and in turn consider these as arrows.

511

@@ -10,36 +16,38 @@ motivation behind the library, the underlying theory, and implementation details

1016This module offers many of the combinators mentioned in the work of

1117[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).

1218

13−See the candidate package in hackage [here](https://hackage.haskell.org/package/laop-0.1.0.0)

19+See the package in hackage [here](https://hackage.haskell.org/package/laop-0.1.1.0)

20+

21+A Functional Pearl has been written and can be regarded as the [reference document](https://github.com/bolt12/tymfgg-pearl) for this library.

1422

1523## Features

1624

17−This library offers:

18−

19−- An internal module representation that uses normal datatypes (`Void`, `()`, `Either`) for matrix dimensions and two

20− type families `FromNat` and `Count` to make it easier to work with these type of matrices.

21−- An interfacing module that uses generalised types for the dimensions and is just a newtype wrapper

22− around the internal matrix data type.

23−- An interfacing module that uses type level naturals for the dimensions and is just a newtype wrapper

24− around the internal matrix data type.

25−- An interfacing module that restricts the matrix contents to being only 0's or 1's (True

26− and False). This module is called 'Relation' and offers many of the combinators seen in

27− AoP and Relational Calculus/Algebra.

28−- A very simple Probabilistic Programming module and several functions and data types to

29− make it easier to deal with sample space.

30−

31−Given this, this new matrix formulation compared to other libraries one has much more advantages, such as:

32−

33−- Has an inductive definition enabling writing matrix manipulation functions in a much more elegant and calculational way;

34−- Statically typed dimensions;

35−- Polymorphic data type dimensions;

36−- Polymorphic matrix content;

37−- Awesome dimension type inference;

38−- Fast type natural conversion via `FromNat` type family;

39−- Matrix `Junc` and `Split`-ing in O(1);

40−- Matrix composition takes advantage of divide-and-conquer and fusion laws.

41−

42−Unfortunately, this approach does not solve the issue of type dimensions being in some way constrainted making it impossible to write Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed.

25+This library offers 3 main matrix programming modules:

26+

27+- One in which matrices are typed with type level natural numbers;

28+- One in which matrices are typed with arbitrary generic data types;

29+- One in which matrices are regarded as relations, i.e. boolean matrices;

30+

31+There's also an experimental module that uses matrices to represent probability

32+distributions.

33+

34+The most interesting feature is that matrices are represented as an inductive data type.

35+Given this formulation, matrix algorithms can be expressed in a much more elegant and

36+calculational style than the traditional vector of vectors representation. This data type

37+guarantees that a matrix will always have valid dimensions. It can also express block matrix

38+computations naturally, which leads to total, efficient and statically typed manipulation

39+and transformation functions.

40+

41+Like matrix multiplication, other common operations, such as matrix transposition, benefit from a block-oriented structure that leads to a simple and natural divide-and-conquer algorithmic solution. Performance wise, this means that without much effort we can obtain optimal cache-oblivious algorithms.

42+

43+Given this, this new matrix formulation compared to other libraries:

44+

45+- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);

46+- Our implementation of matrices enables simple manipulation of submatrices, making it particularly suitable for formal verification and equation reasoning, using the mathematical framework defined by the linear algebra of programming. Furthermore, the data type constructors ensure that the matrices of this kind are sound, i.e. malformed matrices with incorrect dimensions of the sort, can not be constructed.

47+

48+## Known issues

49+

50+Unfortunately, due to the use of type-level programming features, this approach sometimes requires type dimensions to be constrainted, in some way, impossible to write idiomatic Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed. And not all kinds of programs can be modeled using matrices, namely programs that deal with arbitrary infinite data types such as lists or integers (although there are some workarounds).

4351

4452## Notes

4553

Setup.hsdeleted−2

@@ -1,2 +0,0 @@

1−import Distribution.Simple

2−main = defaultMain

app/Main.hsdeleted−7

@@ -1,7 +0,0 @@

1−module Main where

2−

3−import Examples

4−import Benchmarks

5−

6−main :: IO ()

7−main = benchmark

app/Spreadsheets.hsdeleted−61

@@ -1,61 +0,0 @@

1−{-# LANGUAGE DataKinds #-}

2−{-# LANGUAGE GADTs #-}

3−{-# LANGUAGE FlexibleContexts #-}

4−{-# LANGUAGE DeriveGeneric #-}

5−module Spreadsheets

6− (

7− )

8− where

9−

10−import LAoP.Matrix.Type

11−import qualified LAoP.Matrix.Internal as I

12−import GHC.Generics

13−

14−data Student = Student1 | Student2 | Student3 | Student4

15− deriving (Eq, Show, Enum, Bounded, Generic)

16−

17−data Question = Question1 | Question2 | Question3 | Question4

18− deriving (Eq, Show, Enum, Bounded, Generic)

19−

20−data Results = Test | Exam | Final

21− deriving (Eq, Show, Enum, Bounded, Generic)

22−

23−test :: Matrix Float One Results

24−test = point Test

25−

26−exam :: Matrix Float One Results

27−exam = point Exam

28−

29−final :: Matrix Float One Results

30−final = point Final

31−

32−m :: Matrix Float Question Student

33−m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]

34−

35−w :: Matrix Float Question One

36−w = fromLists [[0.2, 0.3, 0.2, 0.3]]

37−

38−xls :: Matrix Float Question One

39− -> Matrix Float Question Student

40− -> Matrix Float Student One

41− -> Matrix Float (Either Question Results) (Either One Student)

42−xls w m t = junc (split w m) (split zeros r)

43− where

44− rExam = m `comp` tr w

45− rTest = tr t

46− rFinal = rTest `maxPP` rExam

47− rAux = junc rTest (junc rExam rFinal)

48− r = tr (converter `comp` tr rAux)

49− converter = junc test (junc exam final)

50−

51−-- | Overloaded, point-wise 'max' function

52−maxPP_ :: Ord e => I.Matrix e a b -> I.Matrix e a b -> I.Matrix e a b

53−maxPP_ I.Empty I.Empty = I.Empty

54−maxPP_ (I.One a) (I.One b) = I.One (a `max` b)

55−maxPP_ (I.Junc a b) (I.Junc c d) = I.Junc (maxPP_ a c) (maxPP_ b d)

56−maxPP_ (I.Split a b) (I.Split c d) = I.Split (maxPP_ a c) (maxPP_ b d)

57−maxPP_ x@(I.Split _ _) y@(I.Junc _ _) = maxPP_ x (I.abideJS y)

58−maxPP_ x@(I.Junc _ _) y@(I.Split _ _) = maxPP_ (I.abideJS x) y

59−

60−maxPP :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b

61−maxPP (M a) (M b) = M (maxPP_ a b)

appveyor.ymlnew file+34

@@ -0,0 +1,34 @@

1+clone_folder: "c:\\WORK"

2+clone_depth: 5

4+# Do not build feature branch with open Pull Requests

5+skip_branch_with_pr: true

7+platform:

8+ - x86_64

10+cache:

11+ - "C:\\SR"

12+ - dist-newstyle

13+

14+environment:

15+ global:

16+ CABOPTS: --store-dir=C:\\SR

17+

18+ matrix:

19+ - GHCVER: 8.6.5

20+

21+install:

22+ - choco source add -n mistuke -s https://www.myget.org/F/mistuke/api/v2

23+ - choco install -y cabal --version 2.4.1.0

24+ - choco install -y ghc --version 8.6.3

25+ - refreshenv

26+

27+before_build:

28+ - cabal --version

29+ - ghc --version

30+ - cabal %CABOPTS% v2-update

31+

32+build_script:

33+ - cabal %CABOPTS% v2-build --enable-tests

34+ - cabal %CABOPTS% v2-test --enable-tests

app/Benchmarks.hs → benchmark/Main.hsrenamed+16−18

@@ -8,38 +8,36 @@

88{-# LANGUAGE AllowAmbiguousTypes #-}

99{-# LANGUAGE ScopedTypeVariables #-}

1010

11−module Benchmarks

12− ( benchmark

13− )

14− where

11+module Main (main) where

1512

1613import LAoP.Dist.Internal

1714import LAoP.Utils

18−import LAoP.Matrix.Type hiding (CountableN, Countable, FromListsN)

19−import qualified LAoP.Matrix.Type as T (FromListsN)

15+import LAoP.Matrix.Type hiding (CountableN, Countable, FLN)

16+import qualified LAoP.Matrix.Type as T (FLN)

2017import Criterion.Main

2118import Control.DeepSeq

2219import Test.QuickCheck

2320import GHC.TypeLits

2421import Data.Proxy

22+import Prelude hiding (id, (.))

2523

2624selectM2 ::

2725 ( Num e,

28− T.FromListsN e b b,

26+ T.FLN b b,

2927 CountableN b

3028 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

31−selectM2 m y = junc y identity `comp` m

29+selectM2 m y = join y iden `comp` m

3230

3331selectD2 ::

34− ( FromListsN b b,

32+ ( FLN b b,

3533 CountableN b

3634 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

37−selectD2 (D d) m = D (junc m identity `comp` d)

35+selectD2 (D d) m = D (join m iden `comp` d)

3836

39−-- Composition with identity

40−compId m = comp identity m

37+-- Composition with iden

38+compId m = comp iden m

4139

42−randomDist :: forall a . (Countable a, FromListsN () a) => Gen (Dist a)

40+randomDist :: forall a . (Countable a, FLN () a) => Gen (Dist a)

4341randomDist = do

4442 let size = fromInteger (natVal (Proxy :: Proxy (Count a)))

4543 l <- vectorOf size (arbitrary :: Gen Prob)

@@ -48,7 +46,7 @@ randomDist = do

4846 m = fromLists lr

4947 return (D m)

5048

51−randomMatrix :: forall a b . (CountableDimensions a b, FromListsN a b) => Gen (Matrix Prob a b)

49+randomMatrix :: forall a b . (CountableDims a b, FLN a b) => Gen (Matrix Prob a b)

5250randomMatrix = do

5351 let cols = fromInteger (natVal (Proxy :: Proxy (Count a)))

5452 rows = fromInteger (natVal (Proxy :: Proxy (Count b)))

@@ -107,7 +105,7 @@ normalize3 as l = let s = sum l

107105 probs = map (/ s) l

108106 in zip as probs

109107

110−randomDistF :: forall a b . (CountableDimensions a b, CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))

108+randomDistF :: forall a b . (CountableDims a b, CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))

111109randomDistF = do

112110 let a = fromInteger (natVal (Proxy :: Proxy (Count a)))

113111 b = fromInteger (natVal (Proxy :: Proxy (Count b)))

@@ -146,8 +144,8 @@ setupEnv2 = do

146144 distList2 <- generate (resize 1 (randomDistF @(Natural 0 100) @(Natural 0 100)))

147145 return (m21, m40, dist, dist2, distList1, distList2)

148146

149−benchmark :: IO ()

150−benchmark = defaultMain [

147+main :: IO ()

148+main = defaultMain [

151149 env setupEnv $ \ ~(m11, m12, m21, m22, m31, m32) -> bgroup "Matrix Composition" [

152150 bgroup "10x10" [

153151 bench "WHNF - 10x10" $ whnf (comp m11) m12

@@ -162,7 +160,7 @@ benchmark = defaultMain [

162160 , bench "NF - 200x200" $ nf (comp m31) m32

163161 ],

164162 bgroup "200x200 - RULES" [

165− bench "No rules - 200x200" $ nf (comp m21) identity

163+ bench "No rules - 200x200" $ nf (comp m21) iden

166164 , bench "Rules - 200x200" $ nf compId m21

167165 ] ],

168166 env setupEnv2 $ \ ~(m21, m40, dist, dist2, dl1, dl2) -> bgroup "Matrix vs List - `select`" [

hackage.shdeleted−10

@@ -1,10 +0,0 @@

1−#!/bin/sh

2−set -e

3−

4−dir=$(mktemp -d dist-docs.XXXXXX)

5−trap 'rm -r "$dir"' EXIT

6−

7−# assumes cabal 2.4 or later

8−cabal v2-haddock --builddir="$dir" --haddock-for-hackage --enable-doc

9−

10−cabal upload -d $dir/*-docs.tar.gz --publish

laop-0.1.0.0-docs.tar.gzdeleted

Binary file not shown.

laop.cabalnew file+89

@@ -0,0 +1,89 @@

1+cabal-version: 2.4

2+name: laop

3+version: 0.1.1.1

4+synopsis: Matrix programming library

5+description: Matrix programming library that uses a type-safe inductive definition

6+ inspired by the Linear Algebra of Programming

7+homepage: https://github.com/bolt12/laop

8+bug-reports: https://github.com/bolt12/laop/issues

9+license: MIT

10+license-file: LICENSE

11+author: Armando Santos

12+maintainer: Armando Santos <armandoifsantos@gmail.com>

13+copyright: 2020 Armando Santos

14+category: Data

15+build-type: Simple

16+extra-doc-files: README.md

17+ CHANGELOG.md

18+tested-with: GHC == 8.6.5

19+ GHC == 8.8.3

20+

21+source-repository head

22+ type: git

23+ location: https://github.com/bolt12/laop.git

24+

25+common common-options

26+ build-depends: base >= 4.12.0.0 && < 5,

27+ deepseq

28+

29+ ghc-options: -Wall

30+ -Wcompat

31+ -Widentities

32+ -Wincomplete-uni-patterns

33+ -Wincomplete-record-updates

34+ if impl(ghc >= 8.0)

35+ ghc-options: -Wredundant-constraints

36+ if impl(ghc >= 8.2)

37+ ghc-options: -fhide-source-paths

38+ if impl(ghc >= 8.4)

39+ ghc-options: -Wmissing-export-lists

40+ -Wpartial-fields

41+ if impl(ghc >= 8.8)

42+ ghc-options: -Wmissing-deriving-strategies

43+

44+ default-language: Haskell2010

45+

46+library

47+ import: common-options

48+ hs-source-dirs: src

49+ exposed-modules: LAoP.Dist

50+ LAoP.Dist.Internal

51+ LAoP.Matrix.Internal

52+ LAoP.Matrix.Nat

53+ LAoP.Matrix.Type

54+ LAoP.Relation

55+ LAoP.Relation.Internal

56+ LAoP.Utils

57+ LAoP.Utils.Internal

58+

59+test-suite laop-test

60+ import: common-options

61+ type: exitcode-stdio-1.0

62+ hs-source-dirs: test

63+ other-modules: Examples.Readme

64+ main-is: Spec.hs

65+ build-depends: laop

66+ ghc-options: -threaded

67+ -rtsopts

68+ -with-rtsopts=-N

69+ -funfolding-use-threshold=16

70+ -fexcess-precision

71+ -optc-O3

72+ -optc-ffast-math

73+

74+benchmark laop-benchmark

75+ import: common-options

76+ type: exitcode-stdio-1.0

77+ hs-source-dirs: benchmark

78+ main-is: Main.hs

79+ build-depends: criterion

80+ , QuickCheck

81+ , deepseq

82+ , laop

83+ ghc-options: -threaded

84+ -rtsopts

85+ -with-rtsopts=-N

86+ -funfolding-use-threshold=16

87+ -fexcess-precision

88+ -optc-O3

89+ -optc-ffast-math

package.yamldeleted−57

@@ -1,57 +0,0 @@

1−name: laop

2−version: 0.1.0.3

3−github: "bolt12/laop"

4−license: BSD3

5−author: "Armando Santos"

6−maintainer: "armandoifsantos@gmail.com"

7−copyright: "MIT"

8−

9−extra-source-files:

10−- README.md

11−- ChangeLog.md

12−

13−# Metadata used when publishing your package

14−# synopsis: Short description of your package

15−category: Data

16−

17−# To avoid duplicated efforts in documentation and dealing with the

18−# complications of embedding Haddock markup inside cabal files, it is

19−# common to point users to the README.md file.

20−description: Please see the README on GitHub at <https://github.com/bolt12/laop#readme>

21−

22−dependencies:

23−- base >= 4.7 && < 5

24−- deepseq

25−

26−library:

27− source-dirs: src

28−

29−executables:

30− laop-exe:

31− main: Main.hs

32− source-dirs: app

33− ghc-options:

34− - -threaded

35− - -rtsopts

36− - -with-rtsopts=-N

37− - -funfolding-use-threshold=16

38− - -fexcess-precision # Good for doubles

39− - -optc-O3 # Good for doubles

40− - -optc-ffast-math # Good for doubles

41− - -O # Optimize

42− dependencies:

43− - laop

44− - criterion

45− - deepseq

46− - QuickCheck

47−

48−tests:

49− laop-test:

50− main: Spec.hs

51− source-dirs: test

52− ghc-options:

53− - -threaded

54− - -rtsopts

55− - -with-rtsopts=-N

56− dependencies:

57− - laop

report-lazy.htmldeleted−1169

@@ -1,1169 +0,0 @@

1−<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">

2−<html>

3− <head>

4− <meta http-equiv="Content-Type" content="text/html; charset=utf-8">

5− <title>criterion report</title>

6− <script language="javascript" type="text/javascript">

7− /*! jQuery v3.3.1 | (c) JS Foundation and other contributors | jquery.org/license */

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xmin=topSentry,ymin=topSentry,xmax=bottomSentry,ymax=bottomSentry;for(j=0;j<points.length;j+=ps){if(points[j]==null)continue;for(m=0;m<ps;++m){val=points[j+m];f=format[m];if(!f||f.autoscale===false||val==fakeInfinity||val==-fakeInfinity)continue;if(f.x){if(val<xmin)xmin=val;if(val>xmax)xmax=val}if(f.y){if(val<ymin)ymin=val;if(val>ymax)ymax=val}}}if(s.bars.show){var delta;switch(s.bars.align){case"left":delta=0;break;case"right":delta=-s.bars.barWidth;break;default:delta=-s.bars.barWidth/2}if(s.bars.horizontal){ymin+=delta;ymax+=delta+s.bars.barWidth}else{xmin+=delta;xmax+=delta+s.bars.barWidth}}updateAxis(s.xaxis,xmin,xmax);updateAxis(s.yaxis,ymin,ymax)}$.each(allAxes(),function(_,axis){if(axis.datamin==topSentry)axis.datamin=null;if(axis.datamax==bottomSentry)axis.datamax=null})}function setupCanvases(){placeholder.css("padding",0).children().filter(function(){return!$(this).hasClass("flot-overlay")&&!$(this).hasClass("flot-base")}).remove();if(placeholder.css("position")=="static")placeholder.css("position","relative");surface=new Canvas("flot-base",placeholder);overlay=new Canvas("flot-overlay",placeholder);ctx=surface.context;octx=overlay.context;eventHolder=$(overlay.element).unbind();var existing=placeholder.data("plot");if(existing){existing.shutdown();overlay.clear()}placeholder.data("plot",plot)}function bindEvents(){if(options.grid.hoverable){eventHolder.mousemove(onMouseMove);eventHolder.bind("mouseleave",onMouseLeave)}if(options.grid.clickable)eventHolder.click(onClick);executeHooks(hooks.bindEvents,[eventHolder])}function shutdown(){if(redrawTimeout)clearTimeout(redrawTimeout);eventHolder.unbind("mousemove",onMouseMove);eventHolder.unbind("mouseleave",onMouseLeave);eventHolder.unbind("click",onClick);executeHooks(hooks.shutdown,[eventHolder])}function setTransformationHelpers(axis){function identity(x){return x}var s,m,t=axis.options.transform||identity,it=axis.options.inverseTransform;if(axis.direction=="x"){s=axis.scale=plotWidth/Math.abs(t(axis.max)-t(axis.min));m=Math.min(t(axis.max),t(axis.min))}else{s=axis.scale=plotHeight/Math.abs(t(axis.max)-t(axis.min));s=-s;m=Math.max(t(axis.max),t(axis.min))}if(t==identity)axis.p2c=function(p){return(p-m)*s};else axis.p2c=function(p){return(t(p)-m)*s};if(!it)axis.c2p=function(c){return m+c/s};else axis.c2p=function(c){return it(m+c/s)}}function measureTickLabels(axis){var opts=axis.options,ticks=axis.ticks||[],labelWidth=opts.labelWidth||0,labelHeight=opts.labelHeight||0,maxWidth=labelWidth||(axis.direction=="x"?Math.floor(surface.width/(ticks.length||1)):null),legacyStyles=axis.direction+"Axis "+axis.direction+axis.n+"Axis",layer="flot-"+axis.direction+"-axis flot-"+axis.direction+axis.n+"-axis "+legacyStyles,font=opts.font||"flot-tick-label tickLabel";for(var i=0;i<ticks.length;++i){var t=ticks[i];if(!t.label)continue;var info=surface.getTextInfo(layer,t.label,font,null,maxWidth);labelWidth=Math.max(labelWidth,info.width);labelHeight=Math.max(labelHeight,info.height)}axis.labelWidth=opts.labelWidth||labelWidth;axis.labelHeight=opts.labelHeight||labelHeight}function allocateAxisBoxFirstPhase(axis){var lw=axis.labelWidth,lh=axis.labelHeight,pos=axis.options.position,isXAxis=axis.direction==="x",tickLength=axis.options.tickLength,axisMargin=options.grid.axisMargin,padding=options.grid.labelMargin,innermost=true,outermost=true,first=true,found=false;$.each(isXAxis?xaxes:yaxes,function(i,a){if(a&&(a.show||a.reserveSpace)){if(a===axis){found=true}else if(a.options.position===pos){if(found){outermost=false}else{innermost=false}}if(!found){first=false}}});if(outermost){axisMargin=0}if(tickLength==null){tickLength=first?"full":5}if(!isNaN(+tickLength))padding+=+tickLength;if(isXAxis){lh+=padding;if(pos=="bottom"){plotOffset.bottom+=lh+axisMargin;axis.box={top:surface.height-plotOffset.bottom,height:lh}}else{axis.box={top:plotOffset.top+axisMargin,height:lh};plotOffset.top+=lh+axisMargin}}else{lw+=padding;if(pos=="left"){axis.box={left:plotOffset.left+axisMargin,width:lw};plotOffset.left+=lw+axisMargin}else{plotOffset.right+=lw+axisMargin;axis.box={left:surface.width-plotOffset.right,width:lw}}}axis.position=pos;axis.tickLength=tickLength;axis.box.padding=padding;axis.innermost=innermost}function allocateAxisBoxSecondPhase(axis){if(axis.direction=="x"){axis.box.left=plotOffset.left-axis.labelWidth/2;axis.box.width=surface.width-plotOffset.left-plotOffset.right+axis.labelWidth}else{axis.box.top=plotOffset.top-axis.labelHeight/2;axis.box.height=surface.height-plotOffset.bottom-plotOffset.top+axis.labelHeight}}function adjustLayoutForThingsStickingOut(){var minMargin=options.grid.minBorderMargin,axis,i;if(minMargin==null){minMargin=0;for(i=0;i<series.length;++i)minMargin=Math.max(minMargin,2*(series[i].points.radius+series[i].points.lineWidth/2))}var margins={left:minMargin,right:minMargin,top:minMargin,bottom:minMargin};$.each(allAxes(),function(_,axis){if(axis.reserveSpace&&axis.ticks&&axis.ticks.length){if(axis.direction==="x"){margins.left=Math.max(margins.left,axis.labelWidth/2);margins.right=Math.max(margins.right,axis.labelWidth/2)}else{margins.bottom=Math.max(margins.bottom,axis.labelHeight/2);margins.top=Math.max(margins.top,axis.labelHeight/2)}}});plotOffset.left=Math.ceil(Math.max(margins.left,plotOffset.left));plotOffset.right=Math.ceil(Math.max(margins.right,plotOffset.right));plotOffset.top=Math.ceil(Math.max(margins.top,plotOffset.top));plotOffset.bottom=Math.ceil(Math.max(margins.bottom,plotOffset.bottom))}function setupGrid(){var i,axes=allAxes(),showGrid=options.grid.show;for(var a in plotOffset){var margin=options.grid.margin||0;plotOffset[a]=typeof margin=="number"?margin:margin[a]||0}executeHooks(hooks.processOffset,[plotOffset]);for(var a in plotOffset){if(typeof options.grid.borderWidth=="object"){plotOffset[a]+=showGrid?options.grid.borderWidth[a]:0}else{plotOffset[a]+=showGrid?options.grid.borderWidth:0}}$.each(axes,function(_,axis){var axisOpts=axis.options;axis.show=axisOpts.show==null?axis.used:axisOpts.show;axis.reserveSpace=axisOpts.reserveSpace==null?axis.show:axisOpts.reserveSpace;setRange(axis)});if(showGrid){var allocatedAxes=$.grep(axes,function(axis){return axis.show||axis.reserveSpace});$.each(allocatedAxes,function(_,axis){setupTickGeneration(axis);setTicks(axis);snapRangeToTicks(axis,axis.ticks);measureTickLabels(axis)});for(i=allocatedAxes.length-1;i>=0;--i)allocateAxisBoxFirstPhase(allocatedAxes[i]);adjustLayoutForThingsStickingOut();$.each(allocatedAxes,function(_,axis){allocateAxisBoxSecondPhase(axis)})}plotWidth=surface.width-plotOffset.left-plotOffset.right;plotHeight=surface.height-plotOffset.bottom-plotOffset.top;$.each(axes,function(_,axis){setTransformationHelpers(axis)});if(showGrid){drawAxisLabels()}insertLegend()}function setRange(axis){var opts=axis.options,min=+(opts.min!=null?opts.min:axis.datamin),max=+(opts.max!=null?opts.max:axis.datamax),delta=max-min;if(delta==0){var widen=max==0?1:.01;if(opts.min==null)min-=widen;if(opts.max==null||opts.min!=null)max+=widen}else{var margin=opts.autoscaleMargin;if(margin!=null){if(opts.min==null){min-=delta*margin;if(min<0&&axis.datamin!=null&&axis.datamin>=0)min=0}if(opts.max==null){max+=delta*margin;if(max>0&&axis.datamax!=null&&axis.datamax<=0)max=0}}}axis.min=min;axis.max=max}function setupTickGeneration(axis){var opts=axis.options;var noTicks;if(typeof opts.ticks=="number"&&opts.ticks>0)noTicks=opts.ticks;else noTicks=.3*Math.sqrt(axis.direction=="x"?surface.width:surface.height);var delta=(axis.max-axis.min)/noTicks,dec=-Math.floor(Math.log(delta)/Math.LN10),maxDec=opts.tickDecimals;if(maxDec!=null&&dec>maxDec){dec=maxDec}var magn=Math.pow(10,-dec),norm=delta/magn,size;if(norm<1.5){size=1}else if(norm<3){size=2;if(norm>2.25&&(maxDec==null||dec+1<=maxDec)){size=2.5;++dec}}else if(norm<7.5){size=5}else{size=10}size*=magn;if(opts.minTickSize!=null&&size<opts.minTickSize){size=opts.minTickSize}axis.delta=delta;axis.tickDecimals=Math.max(0,maxDec!=null?maxDec:dec);axis.tickSize=opts.tickSize||size;if(opts.mode=="time"&&!axis.tickGenerator){throw new Error("Time mode requires the flot.time plugin.")}if(!axis.tickGenerator){axis.tickGenerator=function(axis){var ticks=[],start=floorInBase(axis.min,axis.tickSize),i=0,v=Number.NaN,prev;do{prev=v;v=start+i*axis.tickSize;ticks.push(v);++i}while(v<axis.max&&v!=prev);return ticks};axis.tickFormatter=function(value,axis){var factor=axis.tickDecimals?Math.pow(10,axis.tickDecimals):1;var formatted=""+Math.round(value*factor)/factor;if(axis.tickDecimals!=null){var decimal=formatted.indexOf(".");var precision=decimal==-1?0:formatted.length-decimal-1;if(precision<axis.tickDecimals){return(precision?formatted:formatted+".")+(""+factor).substr(1,axis.tickDecimals-precision)}}return formatted}}if($.isFunction(opts.tickFormatter))axis.tickFormatter=function(v,axis){return""+opts.tickFormatter(v,axis)};if(opts.alignTicksWithAxis!=null){var otherAxis=(axis.direction=="x"?xaxes:yaxes)[opts.alignTicksWithAxis-1];if(otherAxis&&otherAxis.used&&otherAxis!=axis){var niceTicks=axis.tickGenerator(axis);if(niceTicks.length>0){if(opts.min==null)axis.min=Math.min(axis.min,niceTicks[0]);if(opts.max==null&&niceTicks.length>1)axis.max=Math.max(axis.max,niceTicks[niceTicks.length-1])}axis.tickGenerator=function(axis){var ticks=[],v,i;for(i=0;i<otherAxis.ticks.length;++i){v=(otherAxis.ticks[i].v-otherAxis.min)/(otherAxis.max-otherAxis.min);v=axis.min+v*(axis.max-axis.min);ticks.push(v)}return ticks};if(!axis.mode&&opts.tickDecimals==null){var extraDec=Math.max(0,-Math.floor(Math.log(axis.delta)/Math.LN10)+1),ts=axis.tickGenerator(axis);if(!(ts.length>1&&/\..*0$/.test((ts[1]-ts[0]).toFixed(extraDec))))axis.tickDecimals=extraDec}}}}function setTicks(axis){var oticks=axis.options.ticks,ticks=[];if(oticks==null||typeof oticks=="number"&&oticks>0)ticks=axis.tickGenerator(axis);else if(oticks){if($.isFunction(oticks))ticks=oticks(axis);else ticks=oticks}var i,v;axis.ticks=[];for(i=0;i<ticks.length;++i){var label=null;var t=ticks[i];if(typeof t=="object"){v=+t[0];if(t.length>1)label=t[1]}else v=+t;if(label==null)label=axis.tickFormatter(v,axis);if(!isNaN(v))axis.ticks.push({v:v,label:label})}}function snapRangeToTicks(axis,ticks){if(axis.options.autoscaleMargin&&ticks.length>0){if(axis.options.min==null)axis.min=Math.min(axis.min,ticks[0].v);if(axis.options.max==null&&ticks.length>1)axis.max=Math.max(axis.max,ticks[ticks.length-1].v)}}function draw(){surface.clear();executeHooks(hooks.drawBackground,[ctx]);var grid=options.grid;if(grid.show&&grid.backgroundColor)drawBackground();if(grid.show&&!grid.aboveData){drawGrid()}for(var i=0;i<series.length;++i){executeHooks(hooks.drawSeries,[ctx,series[i]]);drawSeries(series[i])}executeHooks(hooks.draw,[ctx]);if(grid.show&&grid.aboveData){drawGrid()}surface.render();triggerRedrawOverlay()}function extractRange(ranges,coord){var axis,from,to,key,axes=allAxes();for(var i=0;i<axes.length;++i){axis=axes[i];if(axis.direction==coord){key=coord+axis.n+"axis";if(!ranges[key]&&axis.n==1)key=coord+"axis";if(ranges[key]){from=ranges[key].from;to=ranges[key].to;break}}}if(!ranges[key]){axis=coord=="x"?xaxes[0]:yaxes[0];from=ranges[coord+"1"];to=ranges[coord+"2"]}if(from!=null&&to!=null&&from>to){var tmp=from;from=to;to=tmp}return{from:from,to:to,axis:axis}}function drawBackground(){ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.fillStyle=getColorOrGradient(options.grid.backgroundColor,plotHeight,0,"rgba(255, 255, 255, 0)");ctx.fillRect(0,0,plotWidth,plotHeight);ctx.restore()}function drawGrid(){var i,axes,bw,bc;ctx.save();ctx.translate(plotOffset.left,plotOffset.top);var markings=options.grid.markings;if(markings){if($.isFunction(markings)){axes=plot.getAxes();axes.xmin=axes.xaxis.min;axes.xmax=axes.xaxis.max;axes.ymin=axes.yaxis.min;axes.ymax=axes.yaxis.max;markings=markings(axes)}for(i=0;i<markings.length;++i){var m=markings[i],xrange=extractRange(m,"x"),yrange=extractRange(m,"y");if(xrange.from==null)xrange.from=xrange.axis.min;if(xrange.to==null)xrange.to=xrange.axis.max;

19−if(yrange.from==null)yrange.from=yrange.axis.min;if(yrange.to==null)yrange.to=yrange.axis.max;if(xrange.to<xrange.axis.min||xrange.from>xrange.axis.max||yrange.to<yrange.axis.min||yrange.from>yrange.axis.max)continue;xrange.from=Math.max(xrange.from,xrange.axis.min);xrange.to=Math.min(xrange.to,xrange.axis.max);yrange.from=Math.max(yrange.from,yrange.axis.min);yrange.to=Math.min(yrange.to,yrange.axis.max);var xequal=xrange.from===xrange.to,yequal=yrange.from===yrange.to;if(xequal&&yequal){continue}xrange.from=Math.floor(xrange.axis.p2c(xrange.from));xrange.to=Math.floor(xrange.axis.p2c(xrange.to));yrange.from=Math.floor(yrange.axis.p2c(yrange.from));yrange.to=Math.floor(yrange.axis.p2c(yrange.to));if(xequal||yequal){var lineWidth=m.lineWidth||options.grid.markingsLineWidth,subPixel=lineWidth%2?.5:0;ctx.beginPath();ctx.strokeStyle=m.color||options.grid.markingsColor;ctx.lineWidth=lineWidth;if(xequal){ctx.moveTo(xrange.to+subPixel,yrange.from);ctx.lineTo(xrange.to+subPixel,yrange.to)}else{ctx.moveTo(xrange.from,yrange.to+subPixel);ctx.lineTo(xrange.to,yrange.to+subPixel)}ctx.stroke()}else{ctx.fillStyle=m.color||options.grid.markingsColor;ctx.fillRect(xrange.from,yrange.to,xrange.to-xrange.from,yrange.from-yrange.to)}}}axes=allAxes();bw=options.grid.borderWidth;for(var j=0;j<axes.length;++j){var axis=axes[j],box=axis.box,t=axis.tickLength,x,y,xoff,yoff;if(!axis.show||axis.ticks.length==0)continue;ctx.lineWidth=1;if(axis.direction=="x"){x=0;if(t=="full")y=axis.position=="top"?0:plotHeight;else y=box.top-plotOffset.top+(axis.position=="top"?box.height:0)}else{y=0;if(t=="full")x=axis.position=="left"?0:plotWidth;else x=box.left-plotOffset.left+(axis.position=="left"?box.width:0)}if(!axis.innermost){ctx.strokeStyle=axis.options.color;ctx.beginPath();xoff=yoff=0;if(axis.direction=="x")xoff=plotWidth+1;else yoff=plotHeight+1;if(ctx.lineWidth==1){if(axis.direction=="x"){y=Math.floor(y)+.5}else{x=Math.floor(x)+.5}}ctx.moveTo(x,y);ctx.lineTo(x+xoff,y+yoff);ctx.stroke()}ctx.strokeStyle=axis.options.tickColor;ctx.beginPath();for(i=0;i<axis.ticks.length;++i){var v=axis.ticks[i].v;xoff=yoff=0;if(isNaN(v)||v<axis.min||v>axis.max||t=="full"&&(typeof bw=="object"&&bw[axis.position]>0||bw>0)&&(v==axis.min||v==axis.max))continue;if(axis.direction=="x"){x=axis.p2c(v);yoff=t=="full"?-plotHeight:t;if(axis.position=="top")yoff=-yoff}else{y=axis.p2c(v);xoff=t=="full"?-plotWidth:t;if(axis.position=="left")xoff=-xoff}if(ctx.lineWidth==1){if(axis.direction=="x")x=Math.floor(x)+.5;else y=Math.floor(y)+.5}ctx.moveTo(x,y);ctx.lineTo(x+xoff,y+yoff)}ctx.stroke()}if(bw){bc=options.grid.borderColor;if(typeof bw=="object"||typeof bc=="object"){if(typeof bw!=="object"){bw={top:bw,right:bw,bottom:bw,left:bw}}if(typeof bc!=="object"){bc={top:bc,right:bc,bottom:bc,left:bc}}if(bw.top>0){ctx.strokeStyle=bc.top;ctx.lineWidth=bw.top;ctx.beginPath();ctx.moveTo(0-bw.left,0-bw.top/2);ctx.lineTo(plotWidth,0-bw.top/2);ctx.stroke()}if(bw.right>0){ctx.strokeStyle=bc.right;ctx.lineWidth=bw.right;ctx.beginPath();ctx.moveTo(plotWidth+bw.right/2,0-bw.top);ctx.lineTo(plotWidth+bw.right/2,plotHeight);ctx.stroke()}if(bw.bottom>0){ctx.strokeStyle=bc.bottom;ctx.lineWidth=bw.bottom;ctx.beginPath();ctx.moveTo(plotWidth+bw.right,plotHeight+bw.bottom/2);ctx.lineTo(0,plotHeight+bw.bottom/2);ctx.stroke()}if(bw.left>0){ctx.strokeStyle=bc.left;ctx.lineWidth=bw.left;ctx.beginPath();ctx.moveTo(0-bw.left/2,plotHeight+bw.bottom);ctx.lineTo(0-bw.left/2,0);ctx.stroke()}}else{ctx.lineWidth=bw;ctx.strokeStyle=options.grid.borderColor;ctx.strokeRect(-bw/2,-bw/2,plotWidth+bw,plotHeight+bw)}}ctx.restore()}function drawAxisLabels(){$.each(allAxes(),function(_,axis){var box=axis.box,legacyStyles=axis.direction+"Axis "+axis.direction+axis.n+"Axis",layer="flot-"+axis.direction+"-axis flot-"+axis.direction+axis.n+"-axis "+legacyStyles,font=axis.options.font||"flot-tick-label tickLabel",tick,x,y,halign,valign;surface.removeText(layer);if(!axis.show||axis.ticks.length==0)return;for(var i=0;i<axis.ticks.length;++i){tick=axis.ticks[i];if(!tick.label||tick.v<axis.min||tick.v>axis.max)continue;if(axis.direction=="x"){halign="center";x=plotOffset.left+axis.p2c(tick.v);if(axis.position=="bottom"){y=box.top+box.padding}else{y=box.top+box.height-box.padding;valign="bottom"}}else{valign="middle";y=plotOffset.top+axis.p2c(tick.v);if(axis.position=="left"){x=box.left+box.width-box.padding;halign="right"}else{x=box.left+box.padding}}surface.addText(layer,x,y,tick.label,font,null,null,halign,valign)}})}function drawSeries(series){if(series.lines.show)drawSeriesLines(series);if(series.bars.show)drawSeriesBars(series);if(series.points.show)drawSeriesPoints(series)}function drawSeriesLines(series){function plotLine(datapoints,xoffset,yoffset,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize,prevx=null,prevy=null;ctx.beginPath();for(var i=ps;i<points.length;i+=ps){var x1=points[i-ps],y1=points[i-ps+1],x2=points[i],y2=points[i+1];if(x1==null||x2==null)continue;if(y1<=y2&&y1<axisy.min){if(y2<axisy.min)continue;x1=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.min}else if(y2<=y1&&y2<axisy.min){if(y1<axisy.min)continue;x2=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.min}if(y1>=y2&&y1>axisy.max){if(y2>axisy.max)continue;x1=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.max}else if(y2>=y1&&y2>axisy.max){if(y1>axisy.max)continue;x2=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.max}if(x1<=x2&&x1<axisx.min){if(x2<axisx.min)continue;y1=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.min}else if(x2<=x1&&x2<axisx.min){if(x1<axisx.min)continue;y2=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.min}if(x1>=x2&&x1>axisx.max){if(x2>axisx.max)continue;y1=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.max}else if(x2>=x1&&x2>axisx.max){if(x1>axisx.max)continue;y2=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.max}if(x1!=prevx||y1!=prevy)ctx.moveTo(axisx.p2c(x1)+xoffset,axisy.p2c(y1)+yoffset);prevx=x2;prevy=y2;ctx.lineTo(axisx.p2c(x2)+xoffset,axisy.p2c(y2)+yoffset)}ctx.stroke()}function plotLineArea(datapoints,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize,bottom=Math.min(Math.max(0,axisy.min),axisy.max),i=0,top,areaOpen=false,ypos=1,segmentStart=0,segmentEnd=0;while(true){if(ps>0&&i>points.length+ps)break;i+=ps;var x1=points[i-ps],y1=points[i-ps+ypos],x2=points[i],y2=points[i+ypos];if(areaOpen){if(ps>0&&x1!=null&&x2==null){segmentEnd=i;ps=-ps;ypos=2;continue}if(ps<0&&i==segmentStart+ps){ctx.fill();areaOpen=false;ps=-ps;ypos=1;i=segmentStart=segmentEnd+ps;continue}}if(x1==null||x2==null)continue;if(x1<=x2&&x1<axisx.min){if(x2<axisx.min)continue;y1=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.min}else if(x2<=x1&&x2<axisx.min){if(x1<axisx.min)continue;y2=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.min}if(x1>=x2&&x1>axisx.max){if(x2>axisx.max)continue;y1=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.max}else if(x2>=x1&&x2>axisx.max){if(x1>axisx.max)continue;y2=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.max}if(!areaOpen){ctx.beginPath();ctx.moveTo(axisx.p2c(x1),axisy.p2c(bottom));areaOpen=true}if(y1>=axisy.max&&y2>=axisy.max){ctx.lineTo(axisx.p2c(x1),axisy.p2c(axisy.max));ctx.lineTo(axisx.p2c(x2),axisy.p2c(axisy.max));continue}else if(y1<=axisy.min&&y2<=axisy.min){ctx.lineTo(axisx.p2c(x1),axisy.p2c(axisy.min));ctx.lineTo(axisx.p2c(x2),axisy.p2c(axisy.min));continue}var x1old=x1,x2old=x2;if(y1<=y2&&y1<axisy.min&&y2>=axisy.min){x1=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.min}else if(y2<=y1&&y2<axisy.min&&y1>=axisy.min){x2=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.min}if(y1>=y2&&y1>axisy.max&&y2<=axisy.max){x1=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.max}else if(y2>=y1&&y2>axisy.max&&y1<=axisy.max){x2=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.max}if(x1!=x1old){ctx.lineTo(axisx.p2c(x1old),axisy.p2c(y1))}ctx.lineTo(axisx.p2c(x1),axisy.p2c(y1));ctx.lineTo(axisx.p2c(x2),axisy.p2c(y2));if(x2!=x2old){ctx.lineTo(axisx.p2c(x2),axisy.p2c(y2));ctx.lineTo(axisx.p2c(x2old),axisy.p2c(y2))}}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.lineJoin="round";var lw=series.lines.lineWidth,sw=series.shadowSize;if(lw>0&&sw>0){ctx.lineWidth=sw;ctx.strokeStyle="rgba(0,0,0,0.1)";var angle=Math.PI/18;plotLine(series.datapoints,Math.sin(angle)*(lw/2+sw/2),Math.cos(angle)*(lw/2+sw/2),series.xaxis,series.yaxis);ctx.lineWidth=sw/2;plotLine(series.datapoints,Math.sin(angle)*(lw/2+sw/4),Math.cos(angle)*(lw/2+sw/4),series.xaxis,series.yaxis)}ctx.lineWidth=lw;ctx.strokeStyle=series.color;var fillStyle=getFillStyle(series.lines,series.color,0,plotHeight);if(fillStyle){ctx.fillStyle=fillStyle;plotLineArea(series.datapoints,series.xaxis,series.yaxis)}if(lw>0)plotLine(series.datapoints,0,0,series.xaxis,series.yaxis);ctx.restore()}function drawSeriesPoints(series){function plotPoints(datapoints,radius,fillStyle,offset,shadow,axisx,axisy,symbol){var points=datapoints.points,ps=datapoints.pointsize;for(var i=0;i<points.length;i+=ps){var x=points[i],y=points[i+1];if(x==null||x<axisx.min||x>axisx.max||y<axisy.min||y>axisy.max)continue;ctx.beginPath();x=axisx.p2c(x);y=axisy.p2c(y)+offset;if(symbol=="circle")ctx.arc(x,y,radius,0,shadow?Math.PI:Math.PI*2,false);else symbol(ctx,x,y,radius,shadow);ctx.closePath();if(fillStyle){ctx.fillStyle=fillStyle;ctx.fill()}ctx.stroke()}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);var lw=series.points.lineWidth,sw=series.shadowSize,radius=series.points.radius,symbol=series.points.symbol;if(lw==0)lw=1e-4;if(lw>0&&sw>0){var w=sw/2;ctx.lineWidth=w;ctx.strokeStyle="rgba(0,0,0,0.1)";plotPoints(series.datapoints,radius,null,w+w/2,true,series.xaxis,series.yaxis,symbol);ctx.strokeStyle="rgba(0,0,0,0.2)";plotPoints(series.datapoints,radius,null,w/2,true,series.xaxis,series.yaxis,symbol)}ctx.lineWidth=lw;ctx.strokeStyle=series.color;plotPoints(series.datapoints,radius,getFillStyle(series.points,series.color),0,false,series.xaxis,series.yaxis,symbol);ctx.restore()}function drawBar(x,y,b,barLeft,barRight,fillStyleCallback,axisx,axisy,c,horizontal,lineWidth){var left,right,bottom,top,drawLeft,drawRight,drawTop,drawBottom,tmp;if(horizontal){drawBottom=drawRight=drawTop=true;drawLeft=false;left=b;right=x;top=y+barLeft;bottom=y+barRight;if(right<left){tmp=right;right=left;left=tmp;drawLeft=true;drawRight=false}}else{drawLeft=drawRight=drawTop=true;drawBottom=false;left=x+barLeft;right=x+barRight;bottom=b;top=y;if(top<bottom){tmp=top;top=bottom;bottom=tmp;drawBottom=true;drawTop=false}}if(right<axisx.min||left>axisx.max||top<axisy.min||bottom>axisy.max)return;if(left<axisx.min){left=axisx.min;drawLeft=false}if(right>axisx.max){right=axisx.max;drawRight=false}if(bottom<axisy.min){bottom=axisy.min;drawBottom=false}if(top>axisy.max){top=axisy.max;drawTop=false}left=axisx.p2c(left);bottom=axisy.p2c(bottom);right=axisx.p2c(right);top=axisy.p2c(top);if(fillStyleCallback){c.fillStyle=fillStyleCallback(bottom,top);c.fillRect(left,top,right-left,bottom-top)}if(lineWidth>0&&(drawLeft||drawRight||drawTop||drawBottom)){c.beginPath();c.moveTo(left,bottom);if(drawLeft)c.lineTo(left,top);else c.moveTo(left,top);if(drawTop)c.lineTo(right,top);else c.moveTo(right,top);if(drawRight)c.lineTo(right,bottom);else c.moveTo(right,bottom);if(drawBottom)c.lineTo(left,bottom);else c.moveTo(left,bottom);c.stroke()}}function drawSeriesBars(series){function plotBars(datapoints,barLeft,barRight,fillStyleCallback,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize;for(var i=0;i<points.length;i+=ps){if(points[i]==null)continue;drawBar(points[i],points[i+1],points[i+2],barLeft,barRight,fillStyleCallback,axisx,axisy,ctx,series.bars.horizontal,series.bars.lineWidth)}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.lineWidth=series.bars.lineWidth;ctx.strokeStyle=series.color;var barLeft;switch(series.bars.align){case"left":barLeft=0;break;case"right":barLeft=-series.bars.barWidth;break;default:barLeft=-series.bars.barWidth/2}var fillStyleCallback=series.bars.fill?function(bottom,top){return getFillStyle(series.bars,series.color,bottom,top)}:null;plotBars(series.datapoints,barLeft,barLeft+series.bars.barWidth,fillStyleCallback,series.xaxis,series.yaxis);ctx.restore()}function getFillStyle(filloptions,seriesColor,bottom,top){var fill=filloptions.fill;if(!fill)return null;if(filloptions.fillColor)return getColorOrGradient(filloptions.fillColor,bottom,top,seriesColor);var c=$.color.parse(seriesColor);c.a=typeof fill=="number"?fill:.4;c.normalize();return c.toString()}function insertLegend(){if(options.legend.container!=null){$(options.legend.container).html("")}else{placeholder.find(".legend").remove()}if(!options.legend.show){return}var fragments=[],entries=[],rowStarted=false,lf=options.legend.labelFormatter,s,label;for(var i=0;i<series.length;++i){s=series[i];if(s.label){label=lf?lf(s.label,s):s.label;if(label){entries.push({label:label,color:s.color})}}}if(options.legend.sorted){if($.isFunction(options.legend.sorted)){entries.sort(options.legend.sorted)}else 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pos="",p=options.legend.position,m=options.legend.margin;if(m[0]==null)m=[m,m];if(p.charAt(0)=="n")pos+="top:"+(m[1]+plotOffset.top)+"px;";else if(p.charAt(0)=="s")pos+="bottom:"+(m[1]+plotOffset.bottom)+"px;";if(p.charAt(1)=="e")pos+="right:"+(m[0]+plotOffset.right)+"px;";else if(p.charAt(1)=="w")pos+="left:"+(m[0]+plotOffset.left)+"px;";var legend=$('<div class="legend">'+table.replace('style="','style="position:absolute;'+pos+";")+"</div>").appendTo(placeholder);if(options.legend.backgroundOpacity!=0){var c=options.legend.backgroundColor;if(c==null){c=options.grid.backgroundColor;if(c&&typeof c=="string")c=$.color.parse(c);else c=$.color.extract(legend,"background-color");c.a=1;c=c.toString()}var div=legend.children();$('<div style="position:absolute;width:'+div.width()+"px;height:"+div.height()+"px;"+pos+"background-color:"+c+';"> </div>').prependTo(legend).css("opacity",options.legend.backgroundOpacity)}}}var highlights=[],redrawTimeout=null;function findNearbyItem(mouseX,mouseY,seriesFilter){var maxDistance=options.grid.mouseActiveRadius,smallestDistance=maxDistance*maxDistance+1,item=null,foundPoint=false,i,j,ps;for(i=series.length-1;i>=0;--i){if(!seriesFilter(series[i]))continue;var s=series[i],axisx=s.xaxis,axisy=s.yaxis,points=s.datapoints.points,mx=axisx.c2p(mouseX),my=axisy.c2p(mouseY),maxx=maxDistance/axisx.scale,maxy=maxDistance/axisy.scale;ps=s.datapoints.pointsize;if(axisx.options.inverseTransform)maxx=Number.MAX_VALUE;if(axisy.options.inverseTransform)maxy=Number.MAX_VALUE;if(s.lines.show||s.points.show){for(j=0;j<points.length;j+=ps){var x=points[j],y=points[j+1];if(x==null)continue;if(x-mx>maxx||x-mx<-maxx||y-my>maxy||y-my<-maxy)continue;var dx=Math.abs(axisx.p2c(x)-mouseX),dy=Math.abs(axisy.p2c(y)-mouseY),dist=dx*dx+dy*dy;if(dist<smallestDistance){smallestDistance=dist;item=[i,j/ps]}}}if(s.bars.show&&!item){var barLeft,barRight;switch(s.bars.align){case"left":barLeft=0;break;case"right":barLeft=-s.bars.barWidth;break;default:barLeft=-s.bars.barWidth/2}barRight=barLeft+s.bars.barWidth;for(j=0;j<points.length;j+=ps){var x=points[j],y=points[j+1],b=points[j+2];if(x==null)continue;if(series[i].bars.horizontal?mx<=Math.max(b,x)&&mx>=Math.min(b,x)&&my>=y+barLeft&&my<=y+barRight:mx>=x+barLeft&&mx<=x+barRight&&my>=Math.min(b,y)&&my<=Math.max(b,y))item=[i,j/ps]}}}if(item){i=item[0];j=item[1];ps=series[i].datapoints.pointsize;return{datapoint:series[i].datapoints.points.slice(j*ps,(j+1)*ps),dataIndex:j,series:series[i],seriesIndex:i}}return null}function onMouseMove(e){if(options.grid.hoverable)triggerClickHoverEvent("plothover",e,function(s){return s["hoverable"]!=false})}function onMouseLeave(e){if(options.grid.hoverable)triggerClickHoverEvent("plothover",e,function(s){return false})}function onClick(e){triggerClickHoverEvent("plotclick",e,function(s){return s["clickable"]!=false})}function triggerClickHoverEvent(eventname,event,seriesFilter){var offset=eventHolder.offset(),canvasX=event.pageX-offset.left-plotOffset.left,canvasY=event.pageY-offset.top-plotOffset.top,pos=canvasToAxisCoords({left:canvasX,top:canvasY});pos.pageX=event.pageX;pos.pageY=event.pageY;var item=findNearbyItem(canvasX,canvasY,seriesFilter);if(item){item.pageX=parseInt(item.series.xaxis.p2c(item.datapoint[0])+offset.left+plotOffset.left,10);item.pageY=parseInt(item.series.yaxis.p2c(item.datapoint[1])+offset.top+plotOffset.top,10)}if(options.grid.autoHighlight){for(var i=0;i<highlights.length;++i){var h=highlights[i];if(h.auto==eventname&&!(item&&h.series==item.series&&h.point[0]==item.datapoint[0]&&h.point[1]==item.datapoint[1]))unhighlight(h.series,h.point)}if(item)highlight(item.series,item.datapoint,eventname)}placeholder.trigger(eventname,[pos,item])}function triggerRedrawOverlay(){var t=options.interaction.redrawOverlayInterval;if(t==-1){drawOverlay();return}if(!redrawTimeout)redrawTimeout=setTimeout(drawOverlay,t)}function drawOverlay(){redrawTimeout=null;octx.save();overlay.clear();octx.translate(plotOffset.left,plotOffset.top);var i,hi;for(i=0;i<highlights.length;++i){hi=highlights[i];if(hi.series.bars.show)drawBarHighlight(hi.series,hi.point);else drawPointHighlight(hi.series,hi.point)}octx.restore();executeHooks(hooks.drawOverlay,[octx])}function highlight(s,point,auto){if(typeof s=="number")s=series[s];if(typeof point=="number"){var ps=s.datapoints.pointsize;point=s.datapoints.points.slice(ps*point,ps*(point+1))}var i=indexOfHighlight(s,point);if(i==-1){highlights.push({series:s,point:point,auto:auto});triggerRedrawOverlay()}else if(!auto)highlights[i].auto=false}function unhighlight(s,point){if(s==null&&point==null){highlights=[];triggerRedrawOverlay();return}if(typeof s=="number")s=series[s];if(typeof point=="number"){var ps=s.datapoints.pointsize;point=s.datapoints.points.slice(ps*point,ps*(point+1))}var i=indexOfHighlight(s,point);if(i!=-1){highlights.splice(i,1);triggerRedrawOverlay()}}function indexOfHighlight(s,p){for(var i=0;i<highlights.length;++i){var h=highlights[i];if(h.series==s&&h.point[0]==p[0]&&h.point[1]==p[1])return i}return-1}function drawPointHighlight(series,point){var x=point[0],y=point[1],axisx=series.xaxis,axisy=series.yaxis,highlightColor=typeof series.highlightColor==="string"?series.highlightColor:$.color.parse(series.color).scale("a",.5).toString();if(x<axisx.min||x>axisx.max||y<axisy.min||y>axisy.max)return;var pointRadius=series.points.radius+series.points.lineWidth/2;octx.lineWidth=pointRadius;octx.strokeStyle=highlightColor;var radius=1.5*pointRadius;x=axisx.p2c(x);y=axisy.p2c(y);octx.beginPath();if(series.points.symbol=="circle")octx.arc(x,y,radius,0,2*Math.PI,false);else series.points.symbol(octx,x,y,radius,false);octx.closePath();octx.stroke()}function drawBarHighlight(series,point){var highlightColor=typeof series.highlightColor==="string"?series.highlightColor:$.color.parse(series.color).scale("a",.5).toString(),fillStyle=highlightColor,barLeft;switch(series.bars.align){case"left":barLeft=0;break;case"right":barLeft=-series.bars.barWidth;break;default:barLeft=-series.bars.barWidth/2}octx.lineWidth=series.bars.lineWidth;octx.strokeStyle=highlightColor;drawBar(point[0],point[1],point[2]||0,barLeft,barLeft+series.bars.barWidth,function(){return fillStyle},series.xaxis,series.yaxis,octx,series.bars.horizontal,series.bars.lineWidth)}function getColorOrGradient(spec,bottom,top,defaultColor){if(typeof spec=="string")return spec;else{var gradient=ctx.createLinearGradient(0,top,0,bottom);for(var i=0,l=spec.colors.length;i<l;++i){var c=spec.colors[i];if(typeof c!="string"){var co=$.color.parse(defaultColor);if(c.brightness!=null)co=co.scale("rgb",c.brightness);if(c.opacity!=null)co.a*=c.opacity;c=co.toString()}gradient.addColorStop(i/(l-1),c)}return gradient}}}$.plot=function(placeholder,data,options){var plot=new Plot($(placeholder),data,options,$.plot.plugins);return plot};$.plot.version="0.8.3";$.plot.plugins=[];$.fn.plot=function(data,options){return this.each(function(){$.plot(this,data,options)})};function floorInBase(n,base){return base*Math.floor(n/base)}})(jQuery);

20− </script>

21− <script language="javascript" type="text/javascript">

22− /* Javascript plotting library for jQuery, version 0.8.3.

23−

25−Licensed under the MIT license.

26−

27−*/

28−(function($){var options={series:{points:{errorbars:null,xerr:{err:"x",show:null,asymmetric:null,upperCap:null,lowerCap:null,color:null,radius:null},yerr:{err:"y",show:null,asymmetric:null,upperCap:null,lowerCap:null,color:null,radius:null}}}};function processRawData(plot,series,data,datapoints){if(!series.points.errorbars)return;var format=[{x:true,number:true,required:true},{y:true,number:true,required:true}];var errors=series.points.errorbars;if(errors=="x"||errors=="xy"){if(series.points.xerr.asymmetric){format.push({x:true,number:true,required:true});format.push({x:true,number:true,required:true})}else format.push({x:true,number:true,required:true})}if(errors=="y"||errors=="xy"){if(series.points.yerr.asymmetric){format.push({y:true,number:true,required:true});format.push({y:true,number:true,required:true})}else format.push({y:true,number:true,required:true})}datapoints.format=format}function parseErrors(series,i){var points=series.datapoints.points;var exl=null,exu=null,eyl=null,eyu=null;var xerr=series.points.xerr,yerr=series.points.yerr;var eb=series.points.errorbars;if(eb=="x"||eb=="xy"){if(xerr.asymmetric){exl=points[i+2];exu=points[i+3];if(eb=="xy")if(yerr.asymmetric){eyl=points[i+4];eyu=points[i+5]}else eyl=points[i+4]}else{exl=points[i+2];if(eb=="xy")if(yerr.asymmetric){eyl=points[i+3];eyu=points[i+4]}else eyl=points[i+3]}}else if(eb=="y")if(yerr.asymmetric){eyl=points[i+2];eyu=points[i+3]}else eyl=points[i+2];if(exu==null)exu=exl;if(eyu==null)eyu=eyl;var errRanges=[exl,exu,eyl,eyu];if(!xerr.show){errRanges[0]=null;errRanges[1]=null}if(!yerr.show){errRanges[2]=null;errRanges[3]=null}return errRanges}function drawSeriesErrors(plot,ctx,s){var points=s.datapoints.points,ps=s.datapoints.pointsize,ax=[s.xaxis,s.yaxis],radius=s.points.radius,err=[s.points.xerr,s.points.yerr];var invertX=false;if(ax[0].p2c(ax[0].max)<ax[0].p2c(ax[0].min)){invertX=true;var tmp=err[0].lowerCap;err[0].lowerCap=err[0].upperCap;err[0].upperCap=tmp}var 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257− <![endif]-->

258− </head>

259− <body>

260− <div id="wrap">

261− <div id="main" class="body">

262− <h1>criterion performance measurements</h1>

263−

264−<h2>overview</h2>

265−

266−<p><a href="#grokularation">want to understand this report?</a></p>

267−

268−<div id="overview" class="ovchart" style="width:900px;height:100px;"></div>

269−

270−<h2><a name="b0">Matrix Composition/10x10/WHNF - 10x10</a></h2>

271− <table width="100%">

272− <tbody>

273− <tr>

274− <td><div id="kde0" class="kdechart"

275− style="width:450px;height:278px;"></div></td>

276− <td><div id="time0" class="timechart"

277− style="width:450px;height:278px;"></div></td>

278−<!--

279− <td><div id="cycle0" class="cyclechart"

280− style="width:300px;height:278px;"></div></td>

281−-->

282− </tr>

283− </tbody>

284− </table>

285−

286− <table>

287− <thead class="analysis">

288− <th></th>

289− <th class="cibound"

290− title="0.95 confidence level">lower bound</th>

291− <th>estimate</th>

292− <th class="cibound"

293− title="0.95 confidence level">upper bound</th>

294− </thead>

295− <tbody>

296− <tr>

297− <td>OLS regression</td>

298− <td><span class="confinterval olstimelb0">xxx</span></td>

299− <td><span class="olstimept0">xxx</span></td>

300− <td><span class="confinterval olstimeub0">xxx</span></td>

301− </tr>

302− <tr>

303− <td>R² goodness-of-fit</td>

304− <td><span class="confinterval olsr2lb0">xxx</span></td>

305− <td><span class="olsr2pt0">xxx</span></td>

306− <td><span class="confinterval olsr2ub0">xxx</span></td>

307− </tr>

308− <tr>

309− <td>Mean execution time</td>

310− <td><span class="confinterval citime">2.429905310206746e-8</span></td>

311− <td><span class="time">2.4514866756029205e-8</span></td>

312− <td><span class="confinterval citime">2.48855951010762e-8</span></td>

313− </tr>

314− <tr>

315− <td>Standard deviation</td>

316− <td><span class="confinterval citime">6.998202238158643e-10</span></td>

317− <td><span class="time">9.647785488002723e-10</span></td>

318− <td><span class="confinterval citime">1.4224955576371737e-9</span></td>

319− </tr>

320− </tbody>

321− </table>

322−

323− <span class="outliers">

324− <p>Outlying measurements have severe

325− (<span class="percent">0.6233390798012226</span>%)

326− effect on estimated standard deviation.</p>

327− </span>

328−<h2><a name="b1">Matrix Composition/10x10/NF - 10x10</a></h2>

329− <table width="100%">

330− <tbody>

331− <tr>

332− <td><div id="kde1" class="kdechart"

333− style="width:450px;height:278px;"></div></td>

334− <td><div id="time1" class="timechart"

335− style="width:450px;height:278px;"></div></td>

336−<!--

337− <td><div id="cycle1" class="cyclechart"

338− style="width:300px;height:278px;"></div></td>

339−-->

340− </tr>

341− </tbody>

342− </table>

343−

344− <table>

345− <thead class="analysis">

346− <th></th>

347− <th class="cibound"

348− title="0.95 confidence level">lower bound</th>

349− <th>estimate</th>

350− <th class="cibound"

351− title="0.95 confidence level">upper bound</th>

352− </thead>

353− <tbody>

354− <tr>

355− <td>OLS regression</td>

356− <td><span class="confinterval olstimelb1">xxx</span></td>

357− <td><span class="olstimept1">xxx</span></td>

358− <td><span class="confinterval olstimeub1">xxx</span></td>

359− </tr>

360− <tr>

361− <td>R² goodness-of-fit</td>

362− <td><span class="confinterval olsr2lb1">xxx</span></td>

363− <td><span class="olsr2pt1">xxx</span></td>

364− <td><span class="confinterval olsr2ub1">xxx</span></td>

365− </tr>

366− <tr>

367− <td>Mean execution time</td>

368− <td><span class="confinterval citime">2.0370099798754226e-4</span></td>

369− <td><span class="time">2.0678688880349948e-4</span></td>

370− <td><span class="confinterval citime">2.120795744857106e-4</span></td>

371− </tr>

372− <tr>

373− <td>Standard deviation</td>

374− <td><span class="confinterval citime">9.393316405757422e-6</span></td>

375− <td><span class="time">1.2679488723877765e-5</span></td>

376− <td><span class="confinterval citime">1.685081532022437e-5</span></td>

377− </tr>

378− </tbody>

379− </table>

380−

381− <span class="outliers">

382− <p>Outlying measurements have severe

383− (<span class="percent">0.5927893628180484</span>%)

384− effect on estimated standard deviation.</p>

385− </span>

386−<h2><a name="b2">Matrix Composition/100x100/WHNF - 100x100</a></h2>

387− <table width="100%">

388− <tbody>

389− <tr>

390− <td><div id="kde2" class="kdechart"

391− style="width:450px;height:278px;"></div></td>

392− <td><div id="time2" class="timechart"

393− style="width:450px;height:278px;"></div></td>

394−<!--

395− <td><div id="cycle2" class="cyclechart"

396− style="width:300px;height:278px;"></div></td>

397−-->

398− </tr>

399− </tbody>

400− </table>

401−

402− <table>

403− <thead class="analysis">

404− <th></th>

405− <th class="cibound"

406− title="0.95 confidence level">lower bound</th>

407− <th>estimate</th>

408− <th class="cibound"

409− title="0.95 confidence level">upper bound</th>

410− </thead>

411− <tbody>

412− <tr>

413− <td>OLS regression</td>

414− <td><span class="confinterval olstimelb2">xxx</span></td>

415− <td><span class="olstimept2">xxx</span></td>

416− <td><span class="confinterval olstimeub2">xxx</span></td>

417− </tr>

418− <tr>

419− <td>R² goodness-of-fit</td>

420− <td><span class="confinterval olsr2lb2">xxx</span></td>

421− <td><span class="olsr2pt2">xxx</span></td>

422− <td><span class="confinterval olsr2ub2">xxx</span></td>

423− </tr>

424− <tr>

425− <td>Mean execution time</td>

426− <td><span class="confinterval citime">2.0050477013786835e-8</span></td>

427− <td><span class="time">2.032500748180748e-8</span></td>

428− <td><span class="confinterval citime">2.076846088643864e-8</span></td>

429− </tr>

430− <tr>

431− <td>Standard deviation</td>

432− <td><span class="confinterval citime">8.13204445187969e-10</span></td>

433− <td><span class="time">1.1748079988373767e-9</span></td>

434− <td><span class="confinterval citime">1.9440744217217704e-9</span></td>

435− </tr>

436− </tbody>

437− </table>

438−

439− <span class="outliers">

440− <p>Outlying measurements have severe

441− (<span class="percent">0.783687156722499</span>%)

442− effect on estimated standard deviation.</p>

443− </span>

444−<h2><a name="b3">Matrix Composition/100x100/NF - 100x100</a></h2>

445− <table width="100%">

446− <tbody>

447− <tr>

448− <td><div id="kde3" class="kdechart"

449− style="width:450px;height:278px;"></div></td>

450− <td><div id="time3" class="timechart"

451− style="width:450px;height:278px;"></div></td>

452−<!--

453− <td><div id="cycle3" class="cyclechart"

454− style="width:300px;height:278px;"></div></td>

455−-->

456− </tr>

457− </tbody>

458− </table>

459−

460− <table>

461− <thead class="analysis">

462− <th></th>

463− <th class="cibound"

464− title="0.95 confidence level">lower bound</th>

465− <th>estimate</th>

466− <th class="cibound"

467− title="0.95 confidence level">upper bound</th>

468− </thead>

469− <tbody>

470− <tr>

471− <td>OLS regression</td>

472− <td><span class="confinterval olstimelb3">xxx</span></td>

473− <td><span class="olstimept3">xxx</span></td>

474− <td><span class="confinterval olstimeub3">xxx</span></td>

475− </tr>

476− <tr>

477− <td>R² goodness-of-fit</td>

478− <td><span class="confinterval olsr2lb3">xxx</span></td>

479− <td><span class="olsr2pt3">xxx</span></td>

480− <td><span class="confinterval olsr2ub3">xxx</span></td>

481− </tr>

482− <tr>

483− <td>Mean execution time</td>

484− <td><span class="confinterval citime">0.3498868480000965</span></td>

485− <td><span class="time">0.3562973993334708</span></td>

486− <td><span class="confinterval citime">0.36268595299998196</span></td>

487− </tr>

488− <tr>

489− <td>Standard deviation</td>

490− <td><span class="confinterval citime">3.523610261477639e-3</span></td>

491− <td><span class="time">7.81042655007339e-3</span></td>

492− <td><span class="confinterval citime">9.18947292446396e-3</span></td>

493− </tr>

494− </tbody>

495− </table>

496−

497− <span class="outliers">

498− <p>Outlying measurements have moderate

499− (<span class="percent">0.1875</span>%)

500− effect on estimated standard deviation.</p>

501− </span>

502−<h2><a name="b4">Matrix Composition/200x200/WHNF - 200x200</a></h2>

503− <table width="100%">

504− <tbody>

505− <tr>

506− <td><div id="kde4" class="kdechart"

507− style="width:450px;height:278px;"></div></td>

508− <td><div id="time4" class="timechart"

509− style="width:450px;height:278px;"></div></td>

510−<!--

511− <td><div id="cycle4" class="cyclechart"

512− style="width:300px;height:278px;"></div></td>

513−-->

514− </tr>

515− </tbody>

516− </table>

517−

518− <table>

519− <thead class="analysis">

520− <th></th>

521− <th class="cibound"

522− title="0.95 confidence level">lower bound</th>

523− <th>estimate</th>

524− <th class="cibound"

525− title="0.95 confidence level">upper bound</th>

526− </thead>

527− <tbody>

528− <tr>

529− <td>OLS regression</td>

530− <td><span class="confinterval olstimelb4">xxx</span></td>

531− <td><span class="olstimept4">xxx</span></td>

532− <td><span class="confinterval olstimeub4">xxx</span></td>

533− </tr>

534− <tr>

535− <td>R² goodness-of-fit</td>

536− <td><span class="confinterval olsr2lb4">xxx</span></td>

537− <td><span class="olsr2pt4">xxx</span></td>

538− <td><span class="confinterval olsr2ub4">xxx</span></td>

539− </tr>

540− <tr>

541− <td>Mean execution time</td>

542− <td><span class="confinterval citime">2.0235240565823596e-8</span></td>

543− <td><span class="time">2.0667543937993357e-8</span></td>

544− <td><span class="confinterval citime">2.143971075518602e-8</span></td>

545− </tr>

546− <tr>

547− <td>Standard deviation</td>

548− <td><span class="confinterval citime">1.1209286317154773e-9</span></td>

549− <td><span class="time">1.8602778611455406e-9</span></td>

550− <td><span class="confinterval citime">2.8888629297529752e-9</span></td>

551− </tr>

552− </tbody>

553− </table>

554−

555− <span class="outliers">

556− <p>Outlying measurements have severe

557− (<span class="percent">0.8983666121212811</span>%)

558− effect on estimated standard deviation.</p>

559− </span>

560−<h2><a name="b5">Matrix Composition/200x200/NF - 200x200</a></h2>

561− <table width="100%">

562− <tbody>

563− <tr>

564− <td><div id="kde5" class="kdechart"

565− style="width:450px;height:278px;"></div></td>

566− <td><div id="time5" class="timechart"

567− style="width:450px;height:278px;"></div></td>

568−<!--

569− <td><div id="cycle5" class="cyclechart"

570− style="width:300px;height:278px;"></div></td>

571−-->

572− </tr>

573− </tbody>

574− </table>

575−

576− <table>

577− <thead class="analysis">

578− <th></th>

579− <th class="cibound"

580− title="0.95 confidence level">lower bound</th>

581− <th>estimate</th>

582− <th class="cibound"

583− title="0.95 confidence level">upper bound</th>

584− </thead>

585− <tbody>

586− <tr>

587− <td>OLS regression</td>

588− <td><span class="confinterval olstimelb5">xxx</span></td>

589− <td><span class="olstimept5">xxx</span></td>

590− <td><span class="confinterval olstimeub5">xxx</span></td>

591− </tr>

592− <tr>

593− <td>R² goodness-of-fit</td>

594− <td><span class="confinterval olsr2lb5">xxx</span></td>

595− <td><span class="olsr2pt5">xxx</span></td>

596− <td><span class="confinterval olsr2ub5">xxx</span></td>

597− </tr>

598− <tr>

599− <td>Mean execution time</td>

600− <td><span class="confinterval citime">4.562200145314364</span></td>

601− <td><span class="time">4.623436544688502</span></td>

602− <td><span class="confinterval citime">4.66968996875039</span></td>

603− </tr>

604− <tr>

605− <td>Standard deviation</td>

606− <td><span class="confinterval citime">3.460169931915702e-2</span></td>

607− <td><span class="time">6.412809734921064e-2</span></td>

608− <td><span class="confinterval citime">8.889305315202672e-2</span></td>

609− </tr>

610− </tbody>

611− </table>

612−

613− <span class="outliers">

614− <p>Outlying measurements have moderate

615− (<span class="percent">0.18749999999999997</span>%)

616− effect on estimated standard deviation.</p>

617− </span>

618−<h2><a name="b6">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/List Distribution - Applicative version</a></h2>

619− <table width="100%">

620− <tbody>

621− <tr>

622− <td><div id="kde6" class="kdechart"

623− style="width:450px;height:278px;"></div></td>

624− <td><div id="time6" class="timechart"

625− style="width:450px;height:278px;"></div></td>

626−<!--

627− <td><div id="cycle6" class="cyclechart"

628− style="width:300px;height:278px;"></div></td>

629−-->

630− </tr>

631− </tbody>

632− </table>

633−

634− <table>

635− <thead class="analysis">

636− <th></th>

637− <th class="cibound"

638− title="0.95 confidence level">lower bound</th>

639− <th>estimate</th>

640− <th class="cibound"

641− title="0.95 confidence level">upper bound</th>

642− </thead>

643− <tbody>

644− <tr>

645− <td>OLS regression</td>

646− <td><span class="confinterval olstimelb6">xxx</span></td>

647− <td><span class="olstimept6">xxx</span></td>

648− <td><span class="confinterval olstimeub6">xxx</span></td>

649− </tr>

650− <tr>

651− <td>R² goodness-of-fit</td>

652− <td><span class="confinterval olsr2lb6">xxx</span></td>

653− <td><span class="olsr2pt6">xxx</span></td>

654− <td><span class="confinterval olsr2ub6">xxx</span></td>

655− </tr>

656− <tr>

657− <td>Mean execution time</td>

658− <td><span class="confinterval citime">2.2228158466256596</span></td>

659− <td><span class="time">2.3105826970010335</span></td>

660− <td><span class="confinterval citime">2.477541572751458</span></td>

661− </tr>

662− <tr>

663− <td>Standard deviation</td>

664− <td><span class="confinterval citime">1.668510235695031e-3</span></td>

665− <td><span class="time">0.16563995052316</span></td>

666− <td><span class="confinterval citime">0.19608795539767002</span></td>

667− </tr>

668− </tbody>

669− </table>

670−

671− <span class="outliers">

672− <p>Outlying measurements have moderate

673− (<span class="percent">0.2024935146554294</span>%)

674− effect on estimated standard deviation.</p>

675− </span>

676−<h2><a name="b7">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Applicative version</a></h2>

677− <table width="100%">

678− <tbody>

679− <tr>

680− <td><div id="kde7" class="kdechart"

681− style="width:450px;height:278px;"></div></td>

682− <td><div id="time7" class="timechart"

683− style="width:450px;height:278px;"></div></td>

684−<!--

685− <td><div id="cycle7" class="cyclechart"

686− style="width:300px;height:278px;"></div></td>

687−-->

688− </tr>

689− </tbody>

690− </table>

691−

692− <table>

693− <thead class="analysis">

694− <th></th>

695− <th class="cibound"

696− title="0.95 confidence level">lower bound</th>

697− <th>estimate</th>

698− <th class="cibound"

699− title="0.95 confidence level">upper bound</th>

700− </thead>

701− <tbody>

702− <tr>

703− <td>OLS regression</td>

704− <td><span class="confinterval olstimelb7">xxx</span></td>

705− <td><span class="olstimept7">xxx</span></td>

706− <td><span class="confinterval olstimeub7">xxx</span></td>

707− </tr>

708− <tr>

709− <td>R² goodness-of-fit</td>

710− <td><span class="confinterval olsr2lb7">xxx</span></td>

711− <td><span class="olsr2pt7">xxx</span></td>

712− <td><span class="confinterval olsr2ub7">xxx</span></td>

713− </tr>

714− <tr>

715− <td>Mean execution time</td>

716− <td><span class="confinterval citime">2.4757740044307665e-3</span></td>

717− <td><span class="time">2.672449312433179e-3</span></td>

718− <td><span class="confinterval citime">3.145392355479385e-3</span></td>

719− </tr>

720− <tr>

721− <td>Standard deviation</td>

722− <td><span class="confinterval citime">4.61354238444537e-4</span></td>

723− <td><span class="time">9.064184723100453e-4</span></td>

724− <td><span class="confinterval citime">1.692293726555292e-3</span></td>

725− </tr>

726− </tbody>

727− </table>

728−

729− <span class="outliers">

730− <p>Outlying measurements have severe

731− (<span class="percent">0.9598911701243743</span>%)

732− effect on estimated standard deviation.</p>

733− </span>

734−<h2><a name="b8">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Selective version</a></h2>

735− <table width="100%">

736− <tbody>

737− <tr>

738− <td><div id="kde8" class="kdechart"

739− style="width:450px;height:278px;"></div></td>

740− <td><div id="time8" class="timechart"

741− style="width:450px;height:278px;"></div></td>

742−<!--

743− <td><div id="cycle8" class="cyclechart"

744− style="width:300px;height:278px;"></div></td>

745−-->

746− </tr>

747− </tbody>

748− </table>

749−

750− <table>

751− <thead class="analysis">

752− <th></th>

753− <th class="cibound"

754− title="0.95 confidence level">lower bound</th>

755− <th>estimate</th>

756− <th class="cibound"

757− title="0.95 confidence level">upper bound</th>

758− </thead>

759− <tbody>

760− <tr>

761− <td>OLS regression</td>

762− <td><span class="confinterval olstimelb8">xxx</span></td>

763− <td><span class="olstimept8">xxx</span></td>

764− <td><span class="confinterval olstimeub8">xxx</span></td>

765− </tr>

766− <tr>

767− <td>R² goodness-of-fit</td>

768− <td><span class="confinterval olsr2lb8">xxx</span></td>

769− <td><span class="olsr2pt8">xxx</span></td>

770− <td><span class="confinterval olsr2ub8">xxx</span></td>

771− </tr>

772− <tr>

773− <td>Mean execution time</td>

774− <td><span class="confinterval citime">1.0094556598949091e-3</span></td>

775− <td><span class="time">1.0459672120901292e-3</span></td>

776− <td><span class="confinterval citime">1.1900093977774448e-3</span></td>

777− </tr>

778− <tr>

779− <td>Standard deviation</td>

780− <td><span class="confinterval citime">4.936441129145525e-5</span></td>

781− <td><span class="time">2.169293235834326e-4</span></td>

782− <td><span class="confinterval citime">4.929920714921142e-4</span></td>

783− </tr>

784− </tbody>

785− </table>

786−

787− <span class="outliers">

788− <p>Outlying measurements have severe

789− (<span class="percent">0.9240968489851639</span>%)

790− effect on estimated standard deviation.</p>

791− </span>

792−<h2><a name="b9">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Applicative version</a></h2>

793− <table width="100%">

794− <tbody>

795− <tr>

796− <td><div id="kde9" class="kdechart"

797− style="width:450px;height:278px;"></div></td>

798− <td><div id="time9" class="timechart"

799− style="width:450px;height:278px;"></div></td>

800−<!--

801− <td><div id="cycle9" class="cyclechart"

802− style="width:300px;height:278px;"></div></td>

803−-->

804− </tr>

805− </tbody>

806− </table>

807−

808− <table>

809− <thead class="analysis">

810− <th></th>

811− <th class="cibound"

812− title="0.95 confidence level">lower bound</th>

813− <th>estimate</th>

814− <th class="cibound"

815− title="0.95 confidence level">upper bound</th>

816− </thead>

817− <tbody>

818− <tr>

819− <td>OLS regression</td>

820− <td><span class="confinterval olstimelb9">xxx</span></td>

821− <td><span class="olstimept9">xxx</span></td>

822− <td><span class="confinterval olstimeub9">xxx</span></td>

823− </tr>

824− <tr>

825− <td>R² goodness-of-fit</td>

826− <td><span class="confinterval olsr2lb9">xxx</span></td>

827− <td><span class="olsr2pt9">xxx</span></td>

828− <td><span class="confinterval olsr2ub9">xxx</span></td>

829− </tr>

830− <tr>

831− <td>Mean execution time</td>

832− <td><span class="confinterval citime">1.1142126609993284</span></td>

833− <td><span class="time">1.1506916786456713</span></td>

834− <td><span class="confinterval citime">1.1642280000833125</span></td>

835− </tr>

836− <tr>

837− <td>Standard deviation</td>

838− <td><span class="confinterval citime">3.115381125098793e-3</span></td>

839− <td><span class="time">2.468474748870533e-2</span></td>

840− <td><span class="confinterval citime">3.126079530343771e-2</span></td>

841− </tr>

842− </tbody>

843− </table>

844−

845− <span class="outliers">

846− <p>Outlying measurements have moderate

847− (<span class="percent">0.1875</span>%)

848− effect on estimated standard deviation.</p>

849− </span>

850−<h2><a name="b10">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Selective version</a></h2>

851− <table width="100%">

852− <tbody>

853− <tr>

854− <td><div id="kde10" class="kdechart"

855− style="width:450px;height:278px;"></div></td>

856− <td><div id="time10" class="timechart"

857− style="width:450px;height:278px;"></div></td>

858−<!--

859− <td><div id="cycle10" class="cyclechart"

860− style="width:300px;height:278px;"></div></td>

861−-->

862− </tr>

863− </tbody>

864− </table>

865−

866− <table>

867− <thead class="analysis">

868− <th></th>

869− <th class="cibound"

870− title="0.95 confidence level">lower bound</th>

871− <th>estimate</th>

872− <th class="cibound"

873− title="0.95 confidence level">upper bound</th>

874− </thead>

875− <tbody>

876− <tr>

877− <td>OLS regression</td>

878− <td><span class="confinterval olstimelb10">xxx</span></td>

879− <td><span class="olstimept10">xxx</span></td>

880− <td><span class="confinterval olstimeub10">xxx</span></td>

881− </tr>

882− <tr>

883− <td>R² goodness-of-fit</td>

884− <td><span class="confinterval olsr2lb10">xxx</span></td>

885− <td><span class="olsr2pt10">xxx</span></td>

886− <td><span class="confinterval olsr2ub10">xxx</span></td>

887− </tr>

888− <tr>

889− <td>Mean execution time</td>

890− <td><span class="confinterval citime">0.530786815458517</span></td>

891− <td><span class="time">0.5400133454575856</span></td>

892− <td><span class="confinterval citime">0.5473636064980383</span></td>

893− </tr>

894− <tr>

895− <td>Standard deviation</td>

896− <td><span class="confinterval citime">4.333057553316104e-3</span></td>

897− <td><span class="time">9.71574644699696e-3</span></td>

898− <td><span class="confinterval citime">1.3200731985635176e-2</span></td>

899− </tr>

900− </tbody>

901− </table>

902−

903− <span class="outliers">

904− <p>Outlying measurements have moderate

905− (<span class="percent">0.18749999999999997</span>%)

906− effect on estimated standard deviation.</p>

907− </span>

908−

909− <h2><a name="grokularation">understanding this report</a></h2>

910−

911− <p>In this report, each function benchmarked by criterion is assigned

912− a section of its own. The charts in each section are active; if

913− you hover your mouse over data points and annotations, you will see

914− more details.</p>

915−

916− <ul>

917− <li>The chart on the left is a

918− <a href="http://en.wikipedia.org/wiki/Kernel_density_estimation">kernel

919− density estimate</a> (also known as a KDE) of time

920− measurements. This graphs the probability of any given time

921− measurement occurring. A spike indicates that a measurement of a

922− particular time occurred; its height indicates how often that

923− measurement was repeated.</li>

924−

925− <li>The chart on the right is the raw data from which the kernel

926− density estimate is built. The <i>x</i> axis indicates the

927− number of loop iterations, while the <i>y</i> axis shows measured

928− execution time for the given number of loop iterations. The

929− line behind the values is the linear regression prediction of

930− execution time for a given number of iterations. Ideally, all

931− measurements will be on (or very near) this line.</li>

932− </ul>

933−

934− <p>Under the charts is a small table.

935− The first two rows are the results of a linear regression run

936− on the measurements displayed in the right-hand chart.</p>

937−

938− <ul>

939− <li><i>OLS regression</i> indicates the

940− time estimated for a single loop iteration using an ordinary

941− least-squares regression model. This number is more accurate

942− than the <i>mean</i> estimate below it, as it more effectively

943− eliminates measurement overhead and other constant factors.</li>

944− <li><i>R² goodness-of-fit</i> is a measure of how

945− accurately the linear regression model fits the observed

946− measurements. If the measurements are not too noisy, R²

947− should lie between 0.99 and 1, indicating an excellent fit. If

948− the number is below 0.99, something is confounding the accuracy

949− of the linear model.</li>

950− <li><i>Mean execution time</i> and <i>standard deviation</i> are

951− statistics calculated from execution time

952− divided by number of iterations.</li>

953− </ul>

954−

955− <p>We use a statistical technique called

956− the <a href="http://en.wikipedia.org/wiki/Bootstrapping_(statistics)">bootstrap</a>

957− to provide confidence intervals on our estimates. The

958− bootstrap-derived upper and lower bounds on estimates let you see

959− how accurate we believe those estimates to be. (Hover the mouse

960− over the table headers to see the confidence levels.)</p>

961−

962− <p>A noisy benchmarking environment can cause some or many

963− measurements to fall far from the mean. These outlying

964− measurements can have a significant inflationary effect on the

965− estimate of the standard deviation. We calculate and display an

966− estimate of the extent to which the standard deviation has been

967− inflated by outliers.</p>

968−

969−<script type="text/javascript">

970−$(function () {

971− function mangulate(rpt) {

972− var measured = function(key) {

973− var idx = rpt.reportKeys.indexOf(key);

974− return rpt.reportMeasured.map(function(r) { return r[idx]; });

975− };

976− var lowerBound = function(est) {

977− return est.estPoint - est.estError.confIntLDX;

978− };

979− var upperBound = function(est) {

980− return est.estPoint + est.estError.confIntUDX;

981− };

982− var number = rpt.reportNumber;

983− var name = rpt.reportName;

984− var mean = rpt.reportAnalysis.anMean.estPoint;

985− var iters = measured("iters");

986− var times = measured("time");

987− var kdetimes = rpt.reportKDEs[0].kdeValues;

988− var kdepdf = rpt.reportKDEs[0].kdePDF;

989−

990− var meanSecs = mean;

991− var units = $.timeUnits(mean);

992− var rgrs = rpt.reportAnalysis.anRegress[0];

993− var scale = units[0];

994− var olsTime = rgrs.regCoeffs.iters;

995− $(".olstimept" + number).text(function() {

996− return $.renderTime(olsTime.estPoint);

997− });

998− $(".olstimelb" + number).text(function() {

999− return $.renderTime(lowerBound(olsTime));

1000− });

1001− $(".olstimeub" + number).text(function() {

1002− return $.renderTime(upperBound(olsTime));

1003− });

1004− $(".olsr2pt" + number).text(function() {

1005− return rgrs.regRSquare.estPoint.toFixed(3);

1006− });

1007− $(".olsr2lb" + number).text(function() {

1008− return lowerBound(rgrs.regRSquare).toFixed(3);

1009− });

1010− $(".olsr2ub" + number).text(function() {

1011− return upperBound(rgrs.regRSquare).toFixed(3);

1012− });

1013− mean *= scale;

1014− kdetimes = $.scaleBy(scale, kdetimes);

1015− var kq = $("#kde" + number);

1016− var k = $.plot(kq,

1017− [{ label: name + " time densities",

1018− data: $.zip(kdetimes, kdepdf),

1019− }],

1020− { xaxis: { tickFormatter: $.unitFormatter(scale) },

1021− yaxis: { ticks: false },

1022− grid: { borderColor: "#777",

1023− hoverable: true, markings: [ { color: '#6fd3fb',

1024− lineWidth: 1.5, xaxis: { from: mean, to: mean } } ] },

1025− });

1026− var o = k.pointOffset({ x: mean, y: 0});

1027− kq.append('<div class="meanlegend" title="' + $.renderTime(meanSecs) +

1028− '" style="position:absolute;left:' + (o.left + 4) +

1029− 'px;bottom:139px;">mean</div>');

1030− $.addTooltip("#kde" + number,

1031− function(secs) { return $.renderTime(secs / scale); });

1032− var timepairs = new Array(times.length);

1033− var lastiter = iters[iters.length-1];

1034− var olspairs = [[0,0], [lastiter, lastiter * scale * olsTime.estPoint]];

1035− for (var i = 0; i < times.length; i++)

1036− timepairs[i] = [iters[i],times[i]*scale];

1037− iterFormatter = function() {

1038− var denom = 0;

1039− return function(iters) {

1040− if (iters == 0)

1041− return '';

1042− if (denom > 0)

1043− return (iters / denom).toFixed();

1044− var power;

1045− if (iters >= 1e9) {

1046− denom = '1e9'; power = '⁹';

1047− }

1048− if (iters >= 1e6) {

1049− denom = '1e6'; power = '⁶';

1050− }

1051− else if (iters >= 1e3) {

1052− denom = '1e3'; power = '³';

1053− }

1054− else denom = 1;

1055− if (denom > 1) {

1056− iters = (iters / denom).toFixed();

1057− iters += '×10' + power + ' iters';

1058− } else {

1059− iters += ' iters';

1060− }

1061− return iters;

1062− };

1063− };

1064− $.plot($("#time" + number),

1065− [{ label: "regression", data: olspairs,

1066− lines: { show: true } },

1067− { label: name + " times", data: timepairs,

1068− points: { show: true } }],

1069− { grid: { borderColor: "#777", hoverable: true },

1070− xaxis: { tickFormatter: iterFormatter() },

1071− yaxis: { tickFormatter: $.unitFormatter(scale) } });

1072− $.addTooltip("#time" + number,

1073− function(iters,secs) {

1074− return ($.renderTime(secs / scale) + ' / ' +

1075− iters.toLocaleString() + ' iters');

1076− });

1077− if (0) {

1078− var cyclepairs = new Array(cycles.length);

1079− for (var i = 0; i < cycles.length; i++)

1080− cyclepairs[i] = [cycles[i],i];

1081− $.plot($("#cycle" + number),

1082− [{ label: name + " cycles",

1083− data: cyclepairs }],

1084− { points: { show: true },

1085− grid: { borderColor: "#777", hoverable: true },

1086− xaxis: { tickFormatter:

1087− function(cycles,axis) { return cycles + ' cycles'; }},

1088− yaxis: { ticks: false },

1089− });

1090− $.addTooltip("#cycles" + number, function(x,y) { return x + ' cycles'; });

1091− }

1092− };

1093− var reports = 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1094− reports.map(mangulate);

1095−

1096− var benches = ["Matrix Composition/10x10/WHNF - 10x10","Matrix Composition/10x10/NF - 10x10","Matrix Composition/100x100/WHNF - 100x100","Matrix Composition/100x100/NF - 100x100","Matrix Composition/200x200/WHNF - 200x200","Matrix Composition/200x200/NF - 200x200","Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/List Distribution - Applicative version","Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Applicative version","Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Selective version","Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Applicative version","Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Selective version",];

1097− var ylabels = [[-0,"<a href=\"#b0\">Matrix Composition/10x10/WHNF - 10x10</a>"],[-1,"<a href=\"#b1\">Matrix Composition/10x10/NF - 10x10</a>"],[-2,"<a href=\"#b2\">Matrix Composition/100x100/WHNF - 100x100</a>"],[-3,"<a href=\"#b3\">Matrix Composition/100x100/NF - 100x100</a>"],[-4,"<a href=\"#b4\">Matrix Composition/200x200/WHNF - 200x200</a>"],[-5,"<a href=\"#b5\">Matrix Composition/200x200/NF - 200x200</a>"],[-6,"<a href=\"#b6\">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/List Distribution - Applicative version</a>"],[-7,"<a href=\"#b7\">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Applicative version</a>"],[-8,"<a href=\"#b8\">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Selective version</a>"],[-9,"<a href=\"#b9\">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Applicative version</a>"],[-10,"<a href=\"#b10\">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Selective version</a>"],];

1098− var means = $.scaleTimes([2.4514866756029205e-8,2.0678688880349948e-4,2.032500748180748e-8,0.3562973993334708,2.0667543937993357e-8,4.623436544688502,2.3105826970010335,2.672449312433179e-3,1.0459672120901292e-3,1.1506916786456713,0.5400133454575856,]);

1099− var stddevs = [9.647785488002723e-10,1.2679488723877765e-5,1.1748079988373767e-9,7.81042655007339e-3,1.8602778611455406e-9,6.412809734921064e-2,0.16563995052316,9.064184723100453e-4,2.169293235834326e-4,2.468474748870533e-2,9.71574644699696e-3,];

1100− stddevs = $.scaleBy(means[1], stddevs);

1101− var xs = [];

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1115− lines: { show: false },

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1124− max_time = Math.max(max_time, means[0][i] + stddevs[i]);

1125− }

1126− var oq = $("#overview");

1127− var max_x = max_time * 1.02;

1128− o = $.plot(oq, xs, { grid: { borderColor: "#777", hoverable: true },

1129− legend: { show: xs.length > 1 },

1130− xaxis: { max: max_x, zoomRange: [false, max_x], panRange: [0, max_x] },

1131− yaxis: { ticks: ylabels, tickColor: '#ffffff', zoomRange: false, panRange: false },

1132− zoom: { interactive: true, amount: 1.05 },

1133− pan: { interactive: true } });

1134− if (benches.length > 3)

1135− o.getPlaceholder().height(28*benches.length);

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1138− o.draw();

1139− $.addTooltip("#overview", function(x,y) { return $.renderTime(x / means[1]); });

1140−});

1141−$(document).ready(function () {

1142− $(".time").text(function(_, text) {

1143− return $.renderTime(text);

1144− });

1145− $(".citime").text(function(_, text) {

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1152−</script>

1153−

1154− </div>

1155− </div>

1156− <div id="footer">

1157− <div class="body">

1158− <div class="footfirst">

1159− <h2>colophon</h2>

1160− <p>This report was created using the

1161− <a href="http://hackage.haskell.org/package/criterion">criterion</a>

1162− benchmark execution and performance analysis tool.</p>

1163− <p>Criterion is developed and maintained

1164− by <a href="http://www.serpentine.com/blog/">Bryan O'Sullivan</a>.</p>

1165− </div>

1166− </div>

1167− </div>

1168− </body>

1169−</html>

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26−

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258− </head>

259− <body>

260− <div id="wrap">

261− <div id="main" class="body">

262− <h1>criterion performance measurements</h1>

263−

264−<h2>overview</h2>

265−

266−<p><a href="#grokularation">want to understand this report?</a></p>

267−

268−<div id="overview" class="ovchart" style="width:900px;height:100px;"></div>

269−

270−<h2><a name="b0">Matrix Composition/10x10/WHNF - 10x10</a></h2>

271− <table width="100%">

272− <tbody>

273− <tr>

274− <td><div id="kde0" class="kdechart"

275− style="width:450px;height:278px;"></div></td>

276− <td><div id="time0" class="timechart"

277− style="width:450px;height:278px;"></div></td>

278−<!--

279− <td><div id="cycle0" class="cyclechart"

280− style="width:300px;height:278px;"></div></td>

281−-->

282− </tr>

283− </tbody>

284− </table>

285−

286− <table>

287− <thead class="analysis">

288− <th></th>

289− <th class="cibound"

290− title="0.95 confidence level">lower bound</th>

291− <th>estimate</th>

292− <th class="cibound"

293− title="0.95 confidence level">upper bound</th>

294− </thead>

295− <tbody>

296− <tr>

297− <td>OLS regression</td>

298− <td><span class="confinterval olstimelb0">xxx</span></td>

299− <td><span class="olstimept0">xxx</span></td>

300− <td><span class="confinterval olstimeub0">xxx</span></td>

301− </tr>

302− <tr>

303− <td>R² goodness-of-fit</td>

304− <td><span class="confinterval olsr2lb0">xxx</span></td>

305− <td><span class="olsr2pt0">xxx</span></td>

306− <td><span class="confinterval olsr2ub0">xxx</span></td>

307− </tr>

308− <tr>

309− <td>Mean execution time</td>

310− <td><span class="confinterval citime">1.2439476132780966e-4</span></td>

311− <td><span class="time">1.2521282696584778e-4</span></td>

312− <td><span class="confinterval citime">1.2627361635087712e-4</span></td>

313− </tr>

314− <tr>

315− <td>Standard deviation</td>

316− <td><span class="confinterval citime">2.3769830641700065e-6</span></td>

317− <td><span class="time">3.0000227210624054e-6</span></td>

318− <td><span class="confinterval citime">3.868635052370606e-6</span></td>

319− </tr>

320− </tbody>

321− </table>

322−

323− <span class="outliers">

324− <p>Outlying measurements have moderate

325− (<span class="percent">0.19510840974237945</span>%)

326− effect on estimated standard deviation.</p>

327− </span>

328−<h2><a name="b1">Matrix Composition/10x10/NF - 10x10</a></h2>

329− <table width="100%">

330− <tbody>

331− <tr>

332− <td><div id="kde1" class="kdechart"

333− style="width:450px;height:278px;"></div></td>

334− <td><div id="time1" class="timechart"

335− style="width:450px;height:278px;"></div></td>

336−<!--

337− <td><div id="cycle1" class="cyclechart"

338− style="width:300px;height:278px;"></div></td>

339−-->

340− </tr>

341− </tbody>

342− </table>

343−

344− <table>

345− <thead class="analysis">

346− <th></th>

347− <th class="cibound"

348− title="0.95 confidence level">lower bound</th>

349− <th>estimate</th>

350− <th class="cibound"

351− title="0.95 confidence level">upper bound</th>

352− </thead>

353− <tbody>

354− <tr>

355− <td>OLS regression</td>

356− <td><span class="confinterval olstimelb1">xxx</span></td>

357− <td><span class="olstimept1">xxx</span></td>

358− <td><span class="confinterval olstimeub1">xxx</span></td>

359− </tr>

360− <tr>

361− <td>R² goodness-of-fit</td>

362− <td><span class="confinterval olsr2lb1">xxx</span></td>

363− <td><span class="olsr2pt1">xxx</span></td>

364− <td><span class="confinterval olsr2ub1">xxx</span></td>

365− </tr>

366− <tr>

367− <td>Mean execution time</td>

368− <td><span class="confinterval citime">1.4537321208713184e-4</span></td>

369− <td><span class="time">1.532024430909428e-4</span></td>

370− <td><span class="confinterval citime">1.6749129455977834e-4</span></td>

371− </tr>

372− <tr>

373− <td>Standard deviation</td>

374− <td><span class="confinterval citime">2.365045193527146e-5</span></td>

375− <td><span class="time">3.649213223188502e-5</span></td>

376− <td><span class="confinterval citime">5.3596936354843454e-5</span></td>

377− </tr>

378− </tbody>

379− </table>

380−

381− <span class="outliers">

382− <p>Outlying measurements have severe

383− (<span class="percent">0.9609582426012205</span>%)

384− effect on estimated standard deviation.</p>

385− </span>

386−<h2><a name="b2">Matrix Composition/100x100/WHNF - 100x100</a></h2>

387− <table width="100%">

388− <tbody>

389− <tr>

390− <td><div id="kde2" class="kdechart"

391− style="width:450px;height:278px;"></div></td>

392− <td><div id="time2" class="timechart"

393− style="width:450px;height:278px;"></div></td>

394−<!--

395− <td><div id="cycle2" class="cyclechart"

396− style="width:300px;height:278px;"></div></td>

397−-->

398− </tr>

399− </tbody>

400− </table>

401−

402− <table>

403− <thead class="analysis">

404− <th></th>

405− <th class="cibound"

406− title="0.95 confidence level">lower bound</th>

407− <th>estimate</th>

408− <th class="cibound"

409− title="0.95 confidence level">upper bound</th>

410− </thead>

411− <tbody>

412− <tr>

413− <td>OLS regression</td>

414− <td><span class="confinterval olstimelb2">xxx</span></td>

415− <td><span class="olstimept2">xxx</span></td>

416− <td><span class="confinterval olstimeub2">xxx</span></td>

417− </tr>

418− <tr>

419− <td>R² goodness-of-fit</td>

420− <td><span class="confinterval olsr2lb2">xxx</span></td>

421− <td><span class="olsr2pt2">xxx</span></td>

422− <td><span class="confinterval olsr2ub2">xxx</span></td>

423− </tr>

424− <tr>

425− <td>Mean execution time</td>

426− <td><span class="confinterval citime">0.11287564945528825</span></td>

427− <td><span class="time">0.11922032795600032</span></td>

428− <td><span class="confinterval citime">0.12600585030653141</span></td>

429− </tr>

430− <tr>

431− <td>Standard deviation</td>

432− <td><span class="confinterval citime">6.3874541360917384e-3</span></td>

433− <td><span class="time">9.77563027930971e-3</span></td>

434− <td><span class="confinterval citime">1.620035460618151e-2</span></td>

435− </tr>

436− </tbody>

437− </table>

438−

439− <span class="outliers">

440− <p>Outlying measurements have moderate

441− (<span class="percent">0.23184395941421526</span>%)

442− effect on estimated standard deviation.</p>

443− </span>

444−<h2><a name="b3">Matrix Composition/100x100/NF - 100x100</a></h2>

445− <table width="100%">

446− <tbody>

447− <tr>

448− <td><div id="kde3" class="kdechart"

449− style="width:450px;height:278px;"></div></td>

450− <td><div id="time3" class="timechart"

451− style="width:450px;height:278px;"></div></td>

452−<!--

453− <td><div id="cycle3" class="cyclechart"

454− style="width:300px;height:278px;"></div></td>

455−-->

456− </tr>

457− </tbody>

458− </table>

459−

460− <table>

461− <thead class="analysis">

462− <th></th>

463− <th class="cibound"

464− title="0.95 confidence level">lower bound</th>

465− <th>estimate</th>

466− <th class="cibound"

467− title="0.95 confidence level">upper bound</th>

468− </thead>

469− <tbody>

470− <tr>

471− <td>OLS regression</td>

472− <td><span class="confinterval olstimelb3">xxx</span></td>

473− <td><span class="olstimept3">xxx</span></td>

474− <td><span class="confinterval olstimeub3">xxx</span></td>

475− </tr>

476− <tr>

477− <td>R² goodness-of-fit</td>

478− <td><span class="confinterval olsr2lb3">xxx</span></td>

479− <td><span class="olsr2pt3">xxx</span></td>

480− <td><span class="confinterval olsr2ub3">xxx</span></td>

481− </tr>

482− <tr>

483− <td>Mean execution time</td>

484− <td><span class="confinterval citime">0.1110982578725974</span></td>

485− <td><span class="time">0.11758616896428131</span></td>

486− <td><span class="confinterval citime">0.12249179825349378</span></td>

487− </tr>

488− <tr>

489− <td>Standard deviation</td>

490− <td><span class="confinterval citime">5.369746750528417e-3</span></td>

491− <td><span class="time">8.150134548496543e-3</span></td>

492− <td><span class="confinterval citime">1.1065658668699107e-2</span></td>

493− </tr>

494− </tbody>

495− </table>

496−

497− <span class="outliers">

498− <p>Outlying measurements have moderate

499− (<span class="percent">0.22459068358328985</span>%)

500− effect on estimated standard deviation.</p>

501− </span>

502−<h2><a name="b4">Matrix Composition/200x200/WHNF - 200x200</a></h2>

503− <table width="100%">

504− <tbody>

505− <tr>

506− <td><div id="kde4" class="kdechart"

507− style="width:450px;height:278px;"></div></td>

508− <td><div id="time4" class="timechart"

509− style="width:450px;height:278px;"></div></td>

510−<!--

511− <td><div id="cycle4" class="cyclechart"

512− style="width:300px;height:278px;"></div></td>

513−-->

514− </tr>

515− </tbody>

516− </table>

517−

518− <table>

519− <thead class="analysis">

520− <th></th>

521− <th class="cibound"

522− title="0.95 confidence level">lower bound</th>

523− <th>estimate</th>

524− <th class="cibound"

525− title="0.95 confidence level">upper bound</th>

526− </thead>

527− <tbody>

528− <tr>

529− <td>OLS regression</td>

530− <td><span class="confinterval olstimelb4">xxx</span></td>

531− <td><span class="olstimept4">xxx</span></td>

532− <td><span class="confinterval olstimeub4">xxx</span></td>

533− </tr>

534− <tr>

535− <td>R² goodness-of-fit</td>

536− <td><span class="confinterval olsr2lb4">xxx</span></td>

537− <td><span class="olsr2pt4">xxx</span></td>

538− <td><span class="confinterval olsr2ub4">xxx</span></td>

539− </tr>

540− <tr>

541− <td>Mean execution time</td>

542− <td><span class="confinterval citime">0.9781591890205164</span></td>

543− <td><span class="time">1.0404128327718354</span></td>

544− <td><span class="confinterval citime">1.0852378857525764</span></td>

545− </tr>

546− <tr>

547− <td>Standard deviation</td>

548− <td><span class="confinterval citime">2.2371087252395228e-2</span></td>

549− <td><span class="time">6.182803786470918e-2</span></td>

550− <td><span class="confinterval citime">8.480026920295321e-2</span></td>

551− </tr>

552− </tbody>

553− </table>

554−

555− <span class="outliers">

556− <p>Outlying measurements have moderate

557− (<span class="percent">0.18749999999999994</span>%)

558− effect on estimated standard deviation.</p>

559− </span>

560−<h2><a name="b5">Matrix Composition/200x200/NF - 200x200</a></h2>

561− <table width="100%">

562− <tbody>

563− <tr>

564− <td><div id="kde5" class="kdechart"

565− style="width:450px;height:278px;"></div></td>

566− <td><div id="time5" class="timechart"

567− style="width:450px;height:278px;"></div></td>

568−<!--

569− <td><div id="cycle5" class="cyclechart"

570− style="width:300px;height:278px;"></div></td>

571−-->

572− </tr>

573− </tbody>

574− </table>

575−

576− <table>

577− <thead class="analysis">

578− <th></th>

579− <th class="cibound"

580− title="0.95 confidence level">lower bound</th>

581− <th>estimate</th>

582− <th class="cibound"

583− title="0.95 confidence level">upper bound</th>

584− </thead>

585− <tbody>

586− <tr>

587− <td>OLS regression</td>

588− <td><span class="confinterval olstimelb5">xxx</span></td>

589− <td><span class="olstimept5">xxx</span></td>

590− <td><span class="confinterval olstimeub5">xxx</span></td>

591− </tr>

592− <tr>

593− <td>R² goodness-of-fit</td>

594− <td><span class="confinterval olsr2lb5">xxx</span></td>

595− <td><span class="olsr2pt5">xxx</span></td>

596− <td><span class="confinterval olsr2ub5">xxx</span></td>

597− </tr>

598− <tr>

599− <td>Mean execution time</td>

600− <td><span class="confinterval citime">1.1011682298167216</span></td>

601− <td><span class="time">1.1075485033964771</span></td>

602− <td><span class="confinterval citime">1.1151673705599023</span></td>

603− </tr>

604− <tr>

605− <td>Standard deviation</td>

606− <td><span class="confinterval citime">2.193747621276998e-3</span></td>

607− <td><span class="time">7.979551817654461e-3</span></td>

608− <td><span class="confinterval citime">1.077654356866516e-2</span></td>

609− </tr>

610− </tbody>

611− </table>

612−

613− <span class="outliers">

614− <p>Outlying measurements have moderate

615− (<span class="percent">0.1875</span>%)

616− effect on estimated standard deviation.</p>

617− </span>

618−<h2><a name="b6">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/List Distribution - Applicative version</a></h2>

619− <table width="100%">

620− <tbody>

621− <tr>

622− <td><div id="kde6" class="kdechart"

623− style="width:450px;height:278px;"></div></td>

624− <td><div id="time6" class="timechart"

625− style="width:450px;height:278px;"></div></td>

626−<!--

627− <td><div id="cycle6" class="cyclechart"

628− style="width:300px;height:278px;"></div></td>

629−-->

630− </tr>

631− </tbody>

632− </table>

633−

634− <table>

635− <thead class="analysis">

636− <th></th>

637− <th class="cibound"

638− title="0.95 confidence level">lower bound</th>

639− <th>estimate</th>

640− <th class="cibound"

641− title="0.95 confidence level">upper bound</th>

642− </thead>

643− <tbody>

644− <tr>

645− <td>OLS regression</td>

646− <td><span class="confinterval olstimelb6">xxx</span></td>

647− <td><span class="olstimept6">xxx</span></td>

648− <td><span class="confinterval olstimeub6">xxx</span></td>

649− </tr>

650− <tr>

651− <td>R² goodness-of-fit</td>

652− <td><span class="confinterval olsr2lb6">xxx</span></td>

653− <td><span class="olsr2pt6">xxx</span></td>

654− <td><span class="confinterval olsr2ub6">xxx</span></td>

655− </tr>

656− <tr>

657− <td>Mean execution time</td>

658− <td><span class="confinterval citime">2.338956891665778</span></td>

659− <td><span class="time">2.3478542657700623</span></td>

660− <td><span class="confinterval citime">2.359088486582247</span></td>

661− </tr>

662− <tr>

663− <td>Standard deviation</td>

664− <td><span class="confinterval citime">3.62336445641734e-3</span></td>

665− <td><span class="time">1.2441745894453989e-2</span></td>

666− <td><span class="confinterval citime">1.67050971444586e-2</span></td>

667− </tr>

668− </tbody>

669− </table>

670−

671− <span class="outliers">

672− <p>Outlying measurements have moderate

673− (<span class="percent">0.18749999999999997</span>%)

674− effect on estimated standard deviation.</p>

675− </span>

676−<h2><a name="b7">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Applicative version</a></h2>

677− <table width="100%">

678− <tbody>

679− <tr>

680− <td><div id="kde7" class="kdechart"

681− style="width:450px;height:278px;"></div></td>

682− <td><div id="time7" class="timechart"

683− style="width:450px;height:278px;"></div></td>

684−<!--

685− <td><div id="cycle7" class="cyclechart"

686− style="width:300px;height:278px;"></div></td>

687−-->

688− </tr>

689− </tbody>

690− </table>

691−

692− <table>

693− <thead class="analysis">

694− <th></th>

695− <th class="cibound"

696− title="0.95 confidence level">lower bound</th>

697− <th>estimate</th>

698− <th class="cibound"

699− title="0.95 confidence level">upper bound</th>

700− </thead>

701− <tbody>

702− <tr>

703− <td>OLS regression</td>

704− <td><span class="confinterval olstimelb7">xxx</span></td>

705− <td><span class="olstimept7">xxx</span></td>

706− <td><span class="confinterval olstimeub7">xxx</span></td>

707− </tr>

708− <tr>

709− <td>R² goodness-of-fit</td>

710− <td><span class="confinterval olsr2lb7">xxx</span></td>

711− <td><span class="olsr2pt7">xxx</span></td>

712− <td><span class="confinterval olsr2ub7">xxx</span></td>

713− </tr>

714− <tr>

715− <td>Mean execution time</td>

716− <td><span class="confinterval citime">1.6422040204675979e-3</span></td>

717− <td><span class="time">1.7931655457283859e-3</span></td>

718− <td><span class="confinterval citime">1.9714955275173672e-3</span></td>

719− </tr>

720− <tr>

721− <td>Standard deviation</td>

722− <td><span class="confinterval citime">3.9388340187426643e-4</span></td>

723− <td><span class="time">5.021999918501304e-4</span></td>

724− <td><span class="confinterval citime">6.587054587998915e-4</span></td>

725− </tr>

726− </tbody>

727− </table>

728−

729− <span class="outliers">

730− <p>Outlying measurements have severe

731− (<span class="percent">0.9470035753933379</span>%)

732− effect on estimated standard deviation.</p>

733− </span>

734−<h2><a name="b8">Matrix vs List - `select`/Distribution `select` - 100+100 / 100x100/Matrix Distribution - Selective version</a></h2>

735− <table width="100%">

736− <tbody>

737− <tr>

738− <td><div id="kde8" class="kdechart"

739− style="width:450px;height:278px;"></div></td>

740− <td><div id="time8" class="timechart"

741− style="width:450px;height:278px;"></div></td>

742−<!--

743− <td><div id="cycle8" class="cyclechart"

744− style="width:300px;height:278px;"></div></td>

745−-->

746− </tr>

747− </tbody>

748− </table>

749−

750− <table>

751− <thead class="analysis">

752− <th></th>

753− <th class="cibound"

754− title="0.95 confidence level">lower bound</th>

755− <th>estimate</th>

756− <th class="cibound"

757− title="0.95 confidence level">upper bound</th>

758− </thead>

759− <tbody>

760− <tr>

761− <td>OLS regression</td>

762− <td><span class="confinterval olstimelb8">xxx</span></td>

763− <td><span class="olstimept8">xxx</span></td>

764− <td><span class="confinterval olstimeub8">xxx</span></td>

765− </tr>

766− <tr>

767− <td>R² goodness-of-fit</td>

768− <td><span class="confinterval olsr2lb8">xxx</span></td>

769− <td><span class="olsr2pt8">xxx</span></td>

770− <td><span class="confinterval olsr2ub8">xxx</span></td>

771− </tr>

772− <tr>

773− <td>Mean execution time</td>

774− <td><span class="confinterval citime">8.252875148785076e-4</span></td>

775− <td><span class="time">9.038536710022285e-4</span></td>

776− <td><span class="confinterval citime">1.0137042058789136e-3</span></td>

777− </tr>

778− <tr>

779− <td>Standard deviation</td>

780− <td><span class="confinterval citime">2.3352589892872938e-4</span></td>

781− <td><span class="time">3.1773226135504875e-4</span></td>

782− <td><span class="confinterval citime">4.273324974311236e-4</span></td>

783− </tr>

784− </tbody>

785− </table>

786−

787− <span class="outliers">

788− <p>Outlying measurements have severe

789− (<span class="percent">0.984823670419889</span>%)

790− effect on estimated standard deviation.</p>

791− </span>

792−<h2><a name="b9">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Applicative version</a></h2>

793− <table width="100%">

794− <tbody>

795− <tr>

796− <td><div id="kde9" class="kdechart"

797− style="width:450px;height:278px;"></div></td>

798− <td><div id="time9" class="timechart"

799− style="width:450px;height:278px;"></div></td>

800−<!--

801− <td><div id="cycle9" class="cyclechart"

802− style="width:300px;height:278px;"></div></td>

803−-->

804− </tr>

805− </tbody>

806− </table>

807−

808− <table>

809− <thead class="analysis">

810− <th></th>

811− <th class="cibound"

812− title="0.95 confidence level">lower bound</th>

813− <th>estimate</th>

814− <th class="cibound"

815− title="0.95 confidence level">upper bound</th>

816− </thead>

817− <tbody>

818− <tr>

819− <td>OLS regression</td>

820− <td><span class="confinterval olstimelb9">xxx</span></td>

821− <td><span class="olstimept9">xxx</span></td>

822− <td><span class="confinterval olstimeub9">xxx</span></td>

823− </tr>

824− <tr>

825− <td>R² goodness-of-fit</td>

826− <td><span class="confinterval olsr2lb9">xxx</span></td>

827− <td><span class="olsr2pt9">xxx</span></td>

828− <td><span class="confinterval olsr2ub9">xxx</span></td>

829− </tr>

830− <tr>

831− <td>Mean execution time</td>

832− <td><span class="confinterval citime">0.20685697058300623</span></td>

833− <td><span class="time">0.20792234497182815</span></td>

834− <td><span class="confinterval citime">0.20898771936065005</span></td>

835− </tr>

836− <tr>

837− <td>Standard deviation</td>

838− <td><span class="confinterval citime">1.11631780898578e-3</span></td>

839− <td><span class="time">1.5104002558460128e-3</span></td>

840− <td><span class="confinterval citime">1.827809373068682e-3</span></td>

841− </tr>

842− </tbody>

843− </table>

844−

845− <span class="outliers">

846− <p>Outlying measurements have moderate

847− (<span class="percent">0.1388888888888886</span>%)

848− effect on estimated standard deviation.</p>

849− </span>

850−<h2><a name="b10">Matrix vs List - `select`/Matrix `select` - 100x(100+100) / 100x100/Selective version</a></h2>

851− <table width="100%">

852− <tbody>

853− <tr>

854− <td><div id="kde10" class="kdechart"

855− style="width:450px;height:278px;"></div></td>

856− <td><div id="time10" class="timechart"

857− style="width:450px;height:278px;"></div></td>

858−<!--

859− <td><div id="cycle10" class="cyclechart"

860− style="width:300px;height:278px;"></div></td>

861−-->

862− </tr>

863− </tbody>

864− </table>

865−

866− <table>

867− <thead class="analysis">

868− <th></th>

869− <th class="cibound"

870− title="0.95 confidence level">lower bound</th>

871− <th>estimate</th>

872− <th class="cibound"

873− title="0.95 confidence level">upper bound</th>

874− </thead>

875− <tbody>

876− <tr>

877− <td>OLS regression</td>

878− <td><span class="confinterval olstimelb10">xxx</span></td>

879− <td><span class="olstimept10">xxx</span></td>

880− <td><span class="confinterval olstimeub10">xxx</span></td>

881− </tr>

882− <tr>

883− <td>R² goodness-of-fit</td>

884− <td><span class="confinterval olsr2lb10">xxx</span></td>

885− <td><span class="olsr2pt10">xxx</span></td>

886− <td><span class="confinterval olsr2ub10">xxx</span></td>

887− </tr>

888− <tr>

889− <td>Mean execution time</td>

890− <td><span class="confinterval citime">0.1040023584088178</span></td>

891− <td><span class="time">0.10482248682469514</span></td>

892− <td><span class="confinterval citime">0.10534815795558282</span></td>

893− </tr>

894− <tr>

895− <td>Standard deviation</td>

896− <td><span class="confinterval citime">5.86339084666882e-4</span></td>

897− <td><span class="time">1.080285904740125e-3</span></td>

898− <td><span class="confinterval citime">1.5634852865472032e-3</span></td>

899− </tr>

900− </tbody>

901− </table>

902−

903− <span class="outliers">

904− <p>Outlying measurements have slight

905− (<span class="percent">9.87654320987654e-2</span>%)

906− effect on estimated standard deviation.</p>

907− </span>

908−

909− <h2><a name="grokularation">understanding this report</a></h2>

910−

911− <p>In this report, each function benchmarked by criterion is assigned

912− a section of its own. The charts in each section are active; if

913− you hover your mouse over data points and annotations, you will see

914− more details.</p>

915−

916− <ul>

917− <li>The chart on the left is a

918− <a href="http://en.wikipedia.org/wiki/Kernel_density_estimation">kernel

919− density estimate</a> (also known as a KDE) of time

920− measurements. This graphs the probability of any given time

921− measurement occurring. A spike indicates that a measurement of a

922− particular time occurred; its height indicates how often that

923− measurement was repeated.</li>

924−

925− <li>The chart on the right is the raw data from which the kernel

926− density estimate is built. The <i>x</i> axis indicates the

927− number of loop iterations, while the <i>y</i> axis shows measured

928− execution time for the given number of loop iterations. The

929− line behind the values is the linear regression prediction of

930− execution time for a given number of iterations. Ideally, all

931− measurements will be on (or very near) this line.</li>

932− </ul>

933−

934− <p>Under the charts is a small table.

935− The first two rows are the results of a linear regression run

936− on the measurements displayed in the right-hand chart.</p>

937−

938− <ul>

939− <li><i>OLS regression</i> indicates the

940− time estimated for a single loop iteration using an ordinary

941− least-squares regression model. This number is more accurate

942− than the <i>mean</i> estimate below it, as it more effectively

943− eliminates measurement overhead and other constant factors.</li>

944− <li><i>R² goodness-of-fit</i> is a measure of how

945− accurately the linear regression model fits the observed

946− measurements. If the measurements are not too noisy, R²

947− should lie between 0.99 and 1, indicating an excellent fit. If

948− the number is below 0.99, something is confounding the accuracy

949− of the linear model.</li>

950− <li><i>Mean execution time</i> and <i>standard deviation</i> are

951− statistics calculated from execution time

952− divided by number of iterations.</li>

953− </ul>

954−

955− <p>We use a statistical technique called

956− the <a href="http://en.wikipedia.org/wiki/Bootstrapping_(statistics)">bootstrap</a>

957− to provide confidence intervals on our estimates. The

958− bootstrap-derived upper and lower bounds on estimates let you see

959− how accurate we believe those estimates to be. (Hover the mouse

960− over the table headers to see the confidence levels.)</p>

961−

962− <p>A noisy benchmarking environment can cause some or many

963− measurements to fall far from the mean. These outlying

964− measurements can have a significant inflationary effect on the

965− estimate of the standard deviation. We calculate and display an

966− estimate of the extent to which the standard deviation has been

967− inflated by outliers.</p>

968−

969−<script type="text/javascript">

970−$(function () {

971− function mangulate(rpt) {

972− var measured = function(key) {

973− var idx = rpt.reportKeys.indexOf(key);

974− return rpt.reportMeasured.map(function(r) { return r[idx]; });

975− };

976− var lowerBound = function(est) {

977− return est.estPoint - est.estError.confIntLDX;

978− };

979− var upperBound = function(est) {

980− return est.estPoint + est.estError.confIntUDX;

981− };

982− var number = rpt.reportNumber;

983− var name = rpt.reportName;

984− var mean = rpt.reportAnalysis.anMean.estPoint;

985− var iters = measured("iters");

986− var times = measured("time");

987− var kdetimes = rpt.reportKDEs[0].kdeValues;

988− var kdepdf = rpt.reportKDEs[0].kdePDF;

989−

990− var meanSecs = mean;

991− var units = $.timeUnits(mean);

992− var rgrs = rpt.reportAnalysis.anRegress[0];

993− var scale = units[0];

994− var olsTime = rgrs.regCoeffs.iters;

995− $(".olstimept" + number).text(function() {

996− return $.renderTime(olsTime.estPoint);

997− });

998− $(".olstimelb" + number).text(function() {

999− return $.renderTime(lowerBound(olsTime));

1000− });

1001− $(".olstimeub" + number).text(function() {

1002− return $.renderTime(upperBound(olsTime));

1003− });

1004− $(".olsr2pt" + number).text(function() {

1005− return rgrs.regRSquare.estPoint.toFixed(3);

1006− });

1007− $(".olsr2lb" + number).text(function() {

1008− return lowerBound(rgrs.regRSquare).toFixed(3);

1009− });

1010− $(".olsr2ub" + number).text(function() {

1011− return upperBound(rgrs.regRSquare).toFixed(3);

1012− });

1013− mean *= scale;

1014− kdetimes = $.scaleBy(scale, kdetimes);

1015− var kq = $("#kde" + number);

1016− var k = $.plot(kq,

1017− [{ label: name + " time densities",

1018− data: $.zip(kdetimes, kdepdf),

1019− }],

1020− { xaxis: { tickFormatter: $.unitFormatter(scale) },

1021− yaxis: { ticks: false },

1022− grid: { borderColor: "#777",

1023− hoverable: true, markings: [ { color: '#6fd3fb',

1024− lineWidth: 1.5, xaxis: { from: mean, to: mean } } ] },

1025− });

1026− var o = k.pointOffset({ x: mean, y: 0});

1027− kq.append('<div class="meanlegend" title="' + $.renderTime(meanSecs) +

1028− '" style="position:absolute;left:' + (o.left + 4) +

1029− 'px;bottom:139px;">mean</div>');

1030− $.addTooltip("#kde" + number,

1031− function(secs) { return $.renderTime(secs / scale); });

1032− var timepairs = new Array(times.length);

1033− var lastiter = iters[iters.length-1];

1034− var olspairs = [[0,0], [lastiter, lastiter * scale * olsTime.estPoint]];

1035− for (var i = 0; i < times.length; i++)

1036− timepairs[i] = [iters[i],times[i]*scale];

1037− iterFormatter = function() {

1038− var denom = 0;

1039− return function(iters) {

1040− if (iters == 0)

1041− return '';

1042− if (denom > 0)

1043− return (iters / denom).toFixed();

1044− var power;

1045− if (iters >= 1e9) {

1046− denom = '1e9'; power = '⁹';

1047− }

1048− if (iters >= 1e6) {

1049− denom = '1e6'; power = '⁶';

1050− }

1051− else if (iters >= 1e3) {

1052− denom = '1e3'; power = '³';

1053− }

1054− else denom = 1;

1055− if (denom > 1) {

1056− iters = (iters / denom).toFixed();

1057− iters += '×10' + power + ' iters';

1058− } else {

1059− iters += ' iters';

1060− }

1061− return iters;

1062− };

1063− };

1064− $.plot($("#time" + number),

1065− [{ label: "regression", data: olspairs,

1066− lines: { show: true } },

1067− { label: name + " times", data: timepairs,

1068− points: { show: true } }],

1069− { grid: { borderColor: "#777", hoverable: true },

1070− xaxis: { tickFormatter: iterFormatter() },

1071− yaxis: { tickFormatter: $.unitFormatter(scale) } });

1072− $.addTooltip("#time" + number,

1073− function(iters,secs) {

1074− return ($.renderTime(secs / scale) + ' / ' +

1075− iters.toLocaleString() + ' iters');

1076− });

1077− if (0) {

1078− var cyclepairs = new Array(cycles.length);

1079− for (var i = 0; i < cycles.length; i++)

1080− cyclepairs[i] = [cycles[i],i];

1081− $.plot($("#cycle" + number),

1082− [{ label: name + " cycles",

1083− data: cyclepairs }],

1084− { points: { show: true },

1085− grid: { borderColor: "#777", hoverable: true },

1086− xaxis: { tickFormatter:

1087− function(cycles,axis) { return cycles + ' cycles'; }},

1088− yaxis: { ticks: false },

1089− });

1090− $.addTooltip("#cycles" + number, function(x,y) { return x + ' cycles'; });

1091− }

1092− };

1093− var reports = 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1094− reports.map(mangulate);

1095−

1098− var means = $.scaleTimes([1.2521282696584778e-4,1.532024430909428e-4,0.11922032795600032,0.11758616896428131,1.0404128327718354,1.1075485033964771,2.3478542657700623,1.7931655457283859e-3,9.038536710022285e-4,0.20792234497182815,0.10482248682469514,]);

1099− var stddevs = [3.0000227210624054e-6,3.649213223188502e-5,9.77563027930971e-3,8.150134548496543e-3,6.182803786470918e-2,7.979551817654461e-3,1.2441745894453989e-2,5.021999918501304e-4,3.1773226135504875e-4,1.5104002558460128e-3,1.080285904740125e-3,];

1100− stddevs = $.scaleBy(means[1], stddevs);

1101− var xs = [];

1102− var prev = null;

1103− var colors = ["#edc240","#afd8f8","#cb4b4b","#4da74d","#9440ed"];

1104− var max_time = 0;

1105− for (var i = 0; i < means[0].length; i++) {

1106− var name = benches[i].split(/\//);

1107− name.pop();

1108− name = name.join('/');

1109− if (name != prev) {

1110− var color = colors[xs.length / 2 % colors.length];

1111− xs.push({ data: [[means[0][i], -i]], label: name, color: color,

1112− bars: { show: true, horizontal: true, barWidth: 0.75,

1113− align: "center" } });

1114− xs.push({ data: [[means[0][i], -i, stddevs[i]]], color: color,

1115− lines: { show: false },

1116− points: { radius: 0, errorbars: "x",

1117− xerr: { show: true, radius: 5,

1118− lowerCap: "-", upperCap: "-" } } });

1119− prev = name;

1120− } else {

1121− xs[xs.length - 2].data.push([means[0][i], -i]);

1122− xs[xs.length - 1].data.push([means[0][i], -i, stddevs[i]]);

1123− }

1124− max_time = Math.max(max_time, means[0][i] + stddevs[i]);

1125− }

1126− var oq = $("#overview");

1127− var max_x = max_time * 1.02;

1128− o = $.plot(oq, xs, { grid: { borderColor: "#777", hoverable: true },

1129− legend: { show: xs.length > 1 },

1130− xaxis: { max: max_x, zoomRange: [false, max_x], panRange: [0, max_x] },

1131− yaxis: { ticks: ylabels, tickColor: '#ffffff', zoomRange: false, panRange: false },

1132− zoom: { interactive: true, amount: 1.05 },

1133− pan: { interactive: true } });

1134− if (benches.length > 3)

1135− o.getPlaceholder().height(28*benches.length);

1136− o.resize();

1137− o.setupGrid();

1138− o.draw();

1139− $.addTooltip("#overview", function(x,y) { return $.renderTime(x / means[1]); });

1140−});

1141−$(document).ready(function () {

1142− $(".time").text(function(_, text) {

1143− return $.renderTime(text);

1144− });

1145− $(".citime").text(function(_, text) {

1146− return $.renderTime(text);

1147− });

1148− $(".percent").text(function(_, text) {

1149− return (text*100).toFixed(1);

1150− });

1151− });

1152−</script>

1153−

1154− </div>

1155− </div>

1156− <div id="footer">

1157− <div class="body">

1158− <div class="footfirst">

1159− <h2>colophon</h2>

1160− <p>This report was created using the

1161− <a href="http://hackage.haskell.org/package/criterion">criterion</a>

1162− benchmark execution and performance analysis tool.</p>

1163− <p>Criterion is developed and maintained

1164− by <a href="http://www.serpentine.com/blog/">Bryan O'Sullivan</a>.</p>

1165− </div>

1166− </div>

1167− </div>

1168− </body>

1169−</html>

report-strict.pdfdeleted

Binary file not shown.

src/LAoP/Dist.hs+2−2

@@ -39,8 +39,8 @@ module LAoP.Dist

3939 -- * Constraint type synonyms

4040 Countable,

4141 CountableN,

42− CountableDimensionsN,

43− FromListsN,

42+ CountableDimsN,

43+ FLN,

4444 Liftable,

4545 TrivialP,

4646

src/LAoP/Dist/Internal.hs+46−45

@@ -16,8 +16,8 @@ module LAoP.Dist.Internal

1616

1717 Countable,

1818 CountableN,

19− CountableDimensionsN,

20− FromListsN,

19+ CountableDimsN,

20+ FLN,

2121 Liftable,

2222 TrivialP,

2323

@@ -43,7 +43,8 @@ module LAoP.Dist.Internal

4343 )

4444 where

4545

46−import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDimensions, CountableN, CountableDimensionsN, Liftable, FromListsN)

46+import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)

47+import Prelude hiding (id, (.))

4748import qualified LAoP.Matrix.Internal as I

4849import LAoP.Utils

4950import GHC.TypeLits

@@ -61,18 +62,18 @@ newtype Dist a = D (Matrix Prob () a)

6162 deriving (Show, Num, Eq, Ord, NFData) via (Matrix Prob () a)

6263

6364-- | Constraint type synonyms to keep the type signatures less convoluted

64−type Countable a = KnownNat (I.Count a)

65−type CountableN a = KnownNat (I.Count (I.Normalize a))

66−type CountableDimensionsN a b = (CountableN a, CountableN b)

67−type FromListsN a b = I.FromLists Prob (I.Normalize a) (I.Normalize b)

68−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)

69−type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

65+type Countable a = KnownNat (I.Count a)

66+type CountableN a = KnownNat (I.Count (I.Normalize a))

67+type CountableDimsN a b = (CountableN a, CountableN b)

68+type FLN a b = I.FL (I.Normalize a) (I.Normalize b)

69+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)

70+type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

7071

7172-- | Functor instance

7273fmapD ::

7374 ( Liftable a b,

74− CountableDimensionsN a b,

75− FromListsN b a

75+ CountableDimsN a b,

76+ FLN b a

7677 )

7778 =>

7879 (a -> b) -> Dist a -> Dist b

@@ -84,17 +85,17 @@ unitD = D (one 1)

8485

8586-- | Applicative/Monoidal instance 'mult' function

8687multD ::

87− ( CountableDimensionsN a b,

88+ ( CountableDimsN a b,

8889 CountableN (a, b),

89− FromListsN (a, b) a,

90− FromListsN (a, b) b,

90+ FLN (a, b) a,

91+ FLN (a, b) b,

9192 TrivialP a b

9293 ) => Dist a -> Dist b -> Dist (a, b)

93−multD (D a) (D b) = D (khatri a b)

94+multD (D a) (D b) = D (kr a b)

9495

9596-- | Selective instance function

9697selectD ::

97− ( FromListsN b b,

98+ ( FLN b b,

9899 CountableN b

99100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

100101selectD (D d) m = D (selectM d m)

@@ -103,31 +104,31 @@ selectD (D d) m = D (selectM d m)

103104-- functions to apply to a given argument;

104105branchD ::

105106 ( Num e,

106− CountableDimensionsN a b,

107− CountableDimensionsN c (Either b c),

108− FromListsN c b,

109− FromListsN a b,

110− FromListsN a a,

111− FromListsN b b,

112− FromListsN c c,

113− FromListsN b a,

114− FromListsN b c,

115− FromListsN (Either b c) b,

116− FromListsN (Either b c) c

107+ CountableDimsN a b,

108+ CountableDimsN c (Either b c),

109+ FLN c b,

110+ FLN a b,

111+ FLN a a,

112+ FLN b b,

113+ FLN c c,

114+ FLN b a,

115+ FLN b c,

116+ FLN (Either b c) b,

117+ FLN (Either b c) c

117118 )

118119 => Dist (Either a b) -> Matrix Prob a c -> Matrix Prob b c -> Dist c

119120branchD x l r = f x `selectD` g l `selectD` r

120121 where

121− f (D m) = D (split (tr i1) (i1 `comp` tr i2) `comp` m)

122+ f (D m) = D (fork (tr i1) (i1 `comp` tr i2) `comp` m)

122123 g m = i2 `comp` m

123124

124125-- | Branch on a Boolean value, skipping unnecessary computations.

125126ifD ::

126− ( CountableDimensionsN a (Either () a),

127− FromListsN a a,

128− FromListsN a (),

129− FromListsN () a,

130− FromListsN (Either () a) a

127+ ( CountableDimsN a (Either () a),

128+ FLN a a,

129+ FLN a (),

130+ FLN () a,

131+ FLN (Either () a) a

131132 )

132133 => Dist Bool -> Dist a -> Dist a -> Dist a

133134ifD x (D t) (D e) = branchD x' t e

@@ -135,7 +136,7 @@ ifD x (D t) (D e) = branchD x' t e

135136 x' = bool (Right ()) (Left ()) `fmapD` x

136137

137138-- | Monad instance 'return' function

138−returnD :: forall a . (Enum a, FromListsN () a, Countable a) => a -> Dist a

139+returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a

139140returnD a = D (col l)

140141 where

141142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))

@@ -150,7 +151,7 @@ bindD (D d) m = D (m `comp` d)

150151(??) ::

151152 ( Enum a,

152153 Countable a,

153− FromListsN () a

154+ FLN () a

154155 ) => (a -> Bool) -> Dist a -> Prob

155156(??) p d =

156157 let l = toValues d

@@ -160,11 +161,11 @@ bindD (D d) m = D (m `comp` d)

160161-- Distribution Construction

161162

162163-- | Constructs a Bernoulli distribution

163−choose :: (FromListsN () a) => Prob -> Dist a

164+choose :: (FLN () a) => Prob -> Dist a

164165choose prob = D (col [prob, 1 - prob])

165166

166167-- | Creates a distribution given a shape function

167−shape :: (FromListsN () a) => (Prob -> Prob) -> [a] -> Dist a

168+shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a

168169shape _ [] = error "Probability.shape: empty list"

169170shape f xs =

170171 let incr = 1 / fromIntegral (length xs - 1)

@@ -172,23 +173,23 @@ shape f xs =

172173 in fromFreqs (zip xs ps)

173174

174175-- | Constructs a Linear distribution

175−linear :: (FromListsN () a) => [a] -> Dist a

176+linear :: (FLN () a) => [a] -> Dist a

176177linear = shape id

177178

178179-- | Constructs an Uniform distribution

179−uniform :: (FromListsN () a) => [a] -> Dist a

180+uniform :: (FLN () a) => [a] -> Dist a

180181uniform = shape (const 1)

181182

182183-- | Constructs an Negative Exponential distribution

183−negExp :: (FromListsN () a) => [a] -> Dist a

184+negExp :: (FLN () a) => [a] -> Dist a

184185negExp = shape (\x -> exp (-x))

185186

186187-- | Constructs an Normal distribution

187−normal :: (FromListsN () a) => [a] -> Dist a

188+normal :: (FLN () a) => [a] -> Dist a

188189normal = shape (normalCurve 0.5 0.5)

189190

190191-- | Transforms a 'Dist' into a list of pairs.

191−toValues :: forall a . (Enum a, Countable a, FromListsN () a) => Dist a -> [(a, Prob)]

192+toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]

192193toValues (D d) =

193194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))

194195 probs = toList d

@@ -196,7 +197,7 @@ toValues (D d) =

196197 in res

197198

198199-- | Pretty a distribution

199−prettyDist :: forall a. (Show a, Enum a, Countable a, FromListsN () a) => Dist a -> String

200+prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String

200201prettyDist d =

201202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)

202203 w = maximum (map (length . show . fst) values)

@@ -208,12 +209,12 @@ prettyDist d =

208209 showR n x = show x ++ " "

209210

210211-- | Pretty Print a distribution

211−prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FromListsN () a) => Dist a -> IO ()

212+prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()

212213prettyPrintDist = putStrLn . prettyDist @a

213214

214215-- Auxiliary functions

215216

216−fromFreqs :: (FromListsN () a) => [(a,Prob)] -> Dist a

217+fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a

217218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))

218219 where q = sum $ map snd xs

219220

src/LAoP/Matrix/Internal.hs+371−400

@@ -30,7 +30,7 @@

3030-- motivation behind the library, the underlying theory, and implementation details.

3131--

3232-- This module offers many of the combinators mentioned in the work of

33−-- Macedo (2012) and Oliveira (2012).

33+-- Macedo (2012) and Oliveira (2012).

3434--

3535-- This is an Internal module and it is no supposed to be imported.

3636--

@@ -43,17 +43,26 @@ module LAoP.Matrix.Internal

4343 --

4444 -- There exists two type families that make it easier to write

4545 -- matrix dimensions: 'FromNat' and 'Count'. This approach

46− -- leads to a very straightforward implementation

47− -- of LAoP combinators.

46+ -- leads to a very straightforward implementation

47+ -- of LAoP combinators.

4848

4949 -- * Type safe matrix representation

5050 Matrix (..),

5151

52+ -- * Constraint type aliases

53+ Countable,

54+ CountableDims,

55+ CountableN,

56+ CountableDimsN,

57+ FL,

58+ FLN,

59+ Liftable,

60+ Trivial,

61+

5262 -- * Primitives

53− empty,

5463 one,

55− junc,

56− split,

64+ join,

65+ fork,

5766

5867 -- * Auxiliary type families

5968 FromNat,

@@ -66,6 +75,7 @@ module LAoP.Matrix.Internal

6675 toLists,

6776 toList,

6877 matrixBuilder,

78+ matrixBuilder',

6979 row,

7080 col,

7181 zeros,

@@ -76,11 +86,17 @@ module LAoP.Matrix.Internal

7686 -- * Misc

7787 -- ** Get dimensions

7888 columns,

89+ columns',

7990 rows,

91+ rows',

8092

8193 -- ** Matrix Transposition

8294 tr,

8395

96+ -- ** Scalar multiplication/division of matrices

97+ (.|),

98+ (./),

99+

84100 -- ** Selective operator

85101 select,

86102 branch,

@@ -89,16 +105,19 @@ module LAoP.Matrix.Internal

89105 cond,

90106

91107 -- ** Matrix "abiding"

92− abideJS,

93− abideSJ,

108+ abideJF,

109+ abideFJ,

110+

111+ -- ** Zip matrices

112+ zipWithM,

94113

95114 -- * Biproduct approach

96− -- ** Split

115+ -- ** Fork

97116 (===),

98117 -- *** Projections

99118 p1,

100119 p2,

101− -- ** Junc

120+ -- ** Join

102121 (|||),

103122 -- *** Injections

104123 i1,

@@ -111,12 +130,13 @@ module LAoP.Matrix.Internal

111130

112131 -- | Note that given the restrictions imposed it is not possible to

113132 -- implement the standard type classes present in standard Haskell.

133+

114134 -- *** Matrix pairing projections

115− kp1,

116− kp2,

135+ fstM,

136+ sndM,

117137

118138 -- *** Matrix pairing

119− khatri,

139+ kr,

120140

121141 -- * Matrix composition and lifting

122142

@@ -124,10 +144,10 @@ module LAoP.Matrix.Internal

124144

125145 -- | Note that given the restrictions imposed it is not possible to

126146 -- implement the standard type classes present in standard Haskell.

127− identity,

147+ iden,

128148 comp,

129− fromF,

130149 fromF',

150+ fromF,

131151

132152 -- * Matrix printing

133153 pretty,

@@ -156,20 +176,20 @@ import Data.Kind

156176import Data.List

157177import Data.Maybe

158178import Data.Proxy

179+import Data.Typeable

159180import Data.Void

160181import GHC.TypeLits

161182import Data.Type.Equality

162183import GHC.Generics

163184import Control.DeepSeq

164−import Control.Category

165−import Prelude hiding ((.))

185+import Prelude hiding (id, (.))

186+import qualified Prelude (id, (.))

166187

167188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

168189data Matrix e cols rows where

169− Empty :: Matrix e Void Void

170190 One :: e -> Matrix e () ()

171− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

172− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

173193

174194deriving instance (Show e) => Show (Matrix e cols rows)

175195

@@ -196,7 +216,6 @@ type family Count (d :: Type) :: Nat where

196216--

197217-- Thanks to Li-Yao Xia this type family is super fast.

198218type family FromNat (n :: Nat) :: Type where

199− FromNat 0 = Void

200219 FromNat 1 = ()

201220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))

202221

@@ -213,115 +232,79 @@ type family Normalize (d :: Type) :: Type where

213232-- | Constraint type synonyms to keep the type signatures less convoluted

214233type Countable a = KnownNat (Count a)

215234type CountableN a = KnownNat (Count (Normalize a))

216−type CountableDimensions a b = (Countable a, Countable b)

217−type CountableDimensionsN a b = (CountableN a, CountableN b)

218−type FromListsN e a b = FromLists e (Normalize a) (Normalize b)

235+type CountableDims a b = (Countable a, Countable b)

236+type CountableDimsN a b = (CountableN a, CountableN b)

237+type FL a b = FromLists a b

238+type FLN a b = FromLists (Normalize a) (Normalize b)

219239type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

220240type Trivial a = FromNat (Count a) ~ a

221241

222−-- | It isn't possible to implement the 'id' function so it's

223−-- implementation is 'undefined'. However 'comp' can be and this partial

224−-- class implementation exists just to make the code more readable.

225−--

226−-- Please use 'identity' instead.

242+-- | It is possible to implement a constrained version of the category type

243+-- class.

227244instance (Num e) => Category (Matrix e) where

228− id = undefined

229− (.) = comp

245+ type Object (Matrix e) a = (FL a a, Countable a)

246+ id = iden

247+ (.) = comp

230248

231249instance NFData e => NFData (Matrix e cols rows) where

232− rnf Empty = ()

233− rnf (One e) = rnf e

234− rnf (Junc a b) = rnf a `seq` rnf b

235− rnf (Split a b) = rnf a `seq` rnf b

250+ rnf (One e) = rnf e

251+ rnf (Join a b) = rnf a `seq` rnf b

252+ rnf (Fork a b) = rnf a `seq` rnf b

236253

237254instance Eq e => Eq (Matrix e cols rows) where

238− Empty == Empty = True

239− (One a) == (One b) = a == b

240− (Junc a b) == (Junc c d) = a == c && b == d

241− (Split a b) == (Split c d) = a == c && b == d

242− (Split (Junc a b) (Junc c d)) == (Junc (Split a' c') (Split b' d')) =

243− (a == a') && (b == b') && (c == c') && (d == d')

244− (Junc (Split a' c') (Split b' d')) == (Split (Junc a b) (Junc c d)) =

245− (a == a') && (b == b') && (c == c') && (d == d')

255+ (One a) == (One b) = a == b

256+ (Join a b) == (Join c d) = a == c && b == d

257+ (Fork a b) == (Fork c d) = a == c && b == d

258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y

259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y

246260

247261instance Num e => Num (Matrix e cols rows) where

262+ a + b = zipWithM (+) a b

263+

264+ a - b = zipWithM (-) a b

265+

266+ a * b = zipWithM (*) a b

248267

249− Empty + Empty = Empty

250− (One a) + (One b) = One (a + b)

251− (Junc a b) + (Junc c d) = Junc (a + c) (b + d)

252− (Split a b) + (Split c d) = Split (a + c) (b + d)

253− (Split (Junc a b) (Junc c d)) + (Junc (Split a' c') (Split b' d')) =

254− Split (Junc (a + a') (b + b')) (Junc (c + c') (d + d'))

255− (Junc (Split a' c') (Split b' d')) + (Split (Junc a b) (Junc c d)) =

256− Split (Junc (a + a') (b + b')) (Junc (c + c') (d + d'))

257−

258− Empty - Empty = Empty

259− (One a) - (One b) = One (a - b)

260− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)

261− (Split a b) - (Split c d) = Split (a - c) (b - d)

262− (Split (Junc a b) (Junc c d)) - (Junc (Split a' c') (Split b' d')) =

263− Split (Junc (a - a') (b - b')) (Junc (c - c') (d - d'))

264− (Junc (Split a' c') (Split b' d')) - (Split (Junc a b) (Junc c d)) =

265− Split (Junc (a - a') (b - b')) (Junc (c - c') (d - d'))

266−

267− Empty * Empty = Empty

268− (One a) * (One b) = One (a * b)

269− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)

270− (Split a b) * (Split c d) = Split (a * c) (b * d)

271− (Split (Junc a b) (Junc c d)) * (Junc (Split a' c') (Split b' d')) =

272− Split (Junc (a * a') (b * b')) (Junc (c * c') (d * d'))

273− (Junc (Split a' c') (Split b' d')) * (Split (Junc a b) (Junc c d)) =

274− Split (Junc (a * a') (b * b')) (Junc (c * c') (d * d'))

275−

276− abs Empty = Empty

277− abs (One a) = One (abs a)

278− abs (Junc a b) = Junc (abs a) (abs b)

279− abs (Split a b) = Split (abs a) (abs b)

280−

281− signum Empty = Empty

282− signum (One a) = One (signum a)

283− signum (Junc a b) = Junc (signum a) (signum b)

284− signum (Split a b) = Split (signum a) (signum b)

268+ abs (One a) = One (abs a)

269+ abs (Join a b) = Join (abs a) (abs b)

270+ abs (Fork a b) = Fork (abs a) (abs b)

271+

272+ signum (One a) = One (signum a)

273+ signum (Join a b) = Join (signum a) (signum b)

274+ signum (Fork a b) = Fork (signum a) (signum b)

285275

286276instance Ord e => Ord (Matrix e cols rows) where

287− Empty <= Empty = True

288− (One a) <= (One b) = a <= b

289− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)

290− (Split a b) <= (Split c d) = (a <= c) && (b <= d)

291− (Split (Junc a b) (Junc c d)) <= (Junc (Split a' c') (Split b' d')) =

292− (a <= a') && (b <= b') && (c <= c') && (d <= d')

293− (Junc (Split a' c') (Split b' d')) <= (Split (Junc a b) (Junc c d)) =

294− (a <= a') && (b <= b') && (c <= c') && (d <= d')

277+ (One a) <= (One b) = a <= b

278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)

279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)

280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y

281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y

295282

296283-- Primitives

297284

298−-- | Empty matrix constructor

299−empty :: Matrix e Void Void

300−empty = Empty

301−

302285-- | Unit matrix constructor

303286one :: e -> Matrix e () ()

304287one = One

305288

306−-- | Matrix 'Junc' constructor

307−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

308−junc = Junc

289+-- | Matrix 'Join' constructor

290+join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

291+join = Join

309292

310293infixl 3 |||

311294

312−-- | Matrix 'Junc' constructor

295+-- | Matrix 'Join' constructor

313296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

314−(|||) = Junc

297+(|||) = Join

315298

316−-- | Matrix 'Split' constructor

317−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

318−split = Split

299+-- | Matrix 'Fork' constructor

300+fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

301+fork = Fork

319302

320303infixl 2 ===

321304

322−-- | Matrix 'Split' constructor

305+-- | Matrix 'Fork' constructor

323306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

324−(===) = Split

307+(===) = Fork

325308

326309-- Construction

327310

@@ -329,84 +312,101 @@ infixl 2 ===

329312--

330313-- Given that it is not possible to branch on types at the term level type

331314-- classes are needed very much like an inductive definition but on types.

332−class FromLists e cols rows where

315+class FromLists cols rows where

333316 -- | Build a matrix out of a list of list of elements. Throws a runtime

334317 -- error if the dimensions do not match.

335318 fromLists :: [[e]] -> Matrix e cols rows

336319

337−instance FromLists e Void Void where

338− fromLists [] = Empty

339− fromLists _ = error "Wrong dimensions"

340−

341−instance {-# OVERLAPPING #-} FromLists e () () where

320+instance FromLists () () where

342321 fromLists [[e]] = One e

343322 fromLists _ = error "Wrong dimensions"

344323

345−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where

346− fromLists [h : t] = Junc (One h) (fromLists [t])

324+instance (FromLists cols ()) => FromLists (Either () cols) () where

325+ fromLists [h : t] = Join (One h) (fromLists [t])

347326 fromLists _ = error "Wrong dimensions"

348327

349−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) () where

350− fromLists [l] =

328+instance {-# OVERLAPPABLE #-}

329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where

330+ fromLists [l] =

351331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

352− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])

332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])

353333 fromLists _ = error "Wrong dimensions"

354334

355−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where

356− fromLists ([h] : t) = Split (One h) (fromLists t)

335+instance (FromLists () rows) => FromLists () (Either () rows) where

336+ fromLists ([h] : t) = Fork (One h) (fromLists t)

357337 fromLists _ = error "Wrong dimensions"

358338

359−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b) where

360− fromLists l@([_] : _) =

339+instance {-# OVERLAPPABLE #-}

340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where

341+ fromLists l@([_] : _) =

361342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

362− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))

343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))

363344 fromLists _ = error "Wrong dimensions"

364345

365−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, Countable c) => FromLists e (Either a b) (Either c d) where

346+instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)

347+ => FromLists (Either a b) (Either c d) where

366348 fromLists l@(h : t) =

367349 let lh = length h

368350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))

369− condition = all (== lh) (map length t)

351+ condition = all ((== lh) . length) t

370352 in if lh > 0 && condition

371− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))

353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))

372354 else error "Not all rows have the same length"

373355

374356-- | Matrix builder function. Constructs a matrix provided with

375−-- a construction function.

376−matrixBuilder ::

357+-- a construction function that operates with indices.

358+matrixBuilder' ::

377359 forall e cols rows.

378− ( FromLists e cols rows,

379− CountableDimensions cols rows

380− ) =>

381− ((Int, Int) -> e) ->

382− Matrix e cols rows

383−matrixBuilder f =

360+ ( FL cols rows,

361+ CountableDims cols rows

362+ ) => ((Int, Int) -> e) -> Matrix e cols rows

363+matrixBuilder' f =

384364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

385365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))

386366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]

387367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions

388368

369+-- | Matrix builder function. Constructs a matrix provided with

370+-- a construction function that operates with arbitrary types.

371+matrixBuilder ::

372+ forall e a b.

373+ ( FLN a b,

374+ Enum a,

375+ Enum b,

376+ Bounded a,

377+ Bounded b,

378+ Eq a,

379+ Countable b

380+ ) => ((a, b) -> e) -> Matrix e (Normalize a) (Normalize b)

381+matrixBuilder f =

382+ let r = fromInteger $ natVal (Proxy :: Proxy (Count b))

383+ positions = [(a, b) | a <- [minBound .. maxBound], b <- [minBound .. maxBound]]

384+ in fromLists . map (map f) . transpose . buildList r $ positions

385+ where

386+ buildList _ [] = []

387+ buildList r l = take r l : buildList r (drop r l)

388+

389389-- | Constructs a column vector matrix

390−col :: (FromLists e () rows) => [e] -> Matrix e () rows

390+col :: (FL () rows) => [e] -> Matrix e () rows

391391col = fromLists . map (: [])

392392

393393-- | Constructs a row vector matrix

394−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()

394+row :: (FL cols ()) => [e] -> Matrix e cols ()

395395row = fromLists . (: [])

396396

397397-- | Lifts functions to matrices with arbitrary dimensions.

398398--

399399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

400400-- types @a@ or @b@ allows.

401−fromF ::

401+fromF' ::

402402 forall a b cols rows e.

403403 ( Liftable e a b,

404− CountableDimensions cols rows,

405− FromLists e rows cols

404+ CountableDims cols rows,

405+ FL rows cols

406406 ) =>

407407 (a -> b) ->

408408 Matrix e cols rows

409−fromF f =

409+fromF' f =

410410 let minA = minBound @a

411411 maxA = maxBound @a

412412 minB = minBound @b

@@ -416,8 +416,8 @@ fromF f =

416416 elementsA = take ccols [minA .. maxA]

417417 elementsB = take rrows [minB .. maxB]

418418 combinations = (,) <$> elementsA <*> elementsB

419− combAp = map snd . sort . map (\(a, b) -> if f a == b

420− then ((fromEnum a, fromEnum b), 1)

419+ combAp = map snd . sort . map (\(a, b) -> if f a == b

420+ then ((fromEnum a, fromEnum b), 1)

421421 else ((fromEnum a, fromEnum b), 0)) $ combinations

422422 mList = buildList combAp rrows

423423 in tr $ fromLists mList

@@ -427,41 +427,23 @@ fromF f =

427427

428428-- | Lifts functions to matrices with dimensions matching @a@ and @b@

429429-- cardinality's.

430−fromF' ::

430+fromF ::

431431 forall a b e.

432432 ( Liftable e a b,

433− CountableDimensionsN a b,

434− FromListsN e b a

433+ CountableDimsN a b,

434+ FLN b a

435435 ) =>

436436 (a -> b) ->

437437 Matrix e (Normalize a) (Normalize b)

438−fromF' f =

439− let minA = minBound @a

440− maxA = maxBound @a

441− minB = minBound @b

442− maxB = maxBound @b

443− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

444− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

445− elementsA = take ccols [minA .. maxA]

446− elementsB = take rrows [minB .. maxB]

447− combinations = (,) <$> elementsA <*> elementsB

448− combAp = map snd . sort . map (\(a, b) -> if f a == b

449− then ((fromEnum a, fromEnum b), 1)

450− else ((fromEnum a, fromEnum b), 0)) $ combinations

451− mList = buildList combAp rrows

452− in tr $ fromLists mList

453− where

454− buildList [] _ = []

455− buildList l r = take r l : buildList (drop r l) r

438+fromF = fromF'

456439

457440-- Conversion

458441

459442-- | Converts a matrix to a list of lists of elements.

460443toLists :: Matrix e cols rows -> [[e]]

461−toLists Empty = []

462444toLists (One e) = [[e]]

463−toLists (Split l r) = toLists l ++ toLists r

464−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)

445+toLists (Fork l r) = toLists l ++ toLists r

446+toLists (Join l r) = zipWith (++) (toLists l) (toLists r)

465447

466448-- | Converts a matrix to a list of elements.

467449toList :: Matrix e cols rows -> [e]

@@ -470,38 +452,38 @@ toList = concat . toLists

470452-- Zeros Matrix

471453

472454-- | The zero matrix. A matrix wholly filled with zeros.

473−zeros :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows

474−zeros = matrixBuilder (const 0)

455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

456+zeros = matrixBuilder' (const 0)

475457

476458-- Ones Matrix

477459

478460-- | The ones matrix. A matrix wholly filled with ones.

479461--

480462-- Also known as T (Top) matrix.

481−ones :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows

482−ones = matrixBuilder (const 1)

463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

464+ones = matrixBuilder' (const 1)

483465

484466-- Const Matrix

485467

486468-- | The constant matrix constructor. A matrix wholly filled with a given

487469-- value.

488−constant :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => e -> Matrix e cols rows

489−constant e = matrixBuilder (const e)

470+constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows

471+constant e = matrixBuilder' (const e)

490472

491473-- Bang Matrix

492474

493475-- | The T (Top) row vector matrix.

494−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), Countable cols) => Matrix e cols ()

476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()

495477bang =

496478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

497479 in fromLists [take c [1, 1 ..]]

498480

499−-- Identity Matrix

481+-- iden Matrix

500482

501−-- | Identity matrix.

502−identity :: (Num e, FromLists e cols cols, Countable cols) => Matrix e cols cols

503−identity = matrixBuilder (bool 0 1 . uncurry (==))

504−{-# NOINLINE identity #-}

483+-- | iden matrix.

484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols

485+iden = matrixBuilder' (bool 0 1 . uncurry (==))

486+{-# NOINLINE iden #-}

505487

506488-- Matrix composition (MMM)

507489

@@ -510,65 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))

510492-- This definition takes advantage of divide-and-conquer and fusion laws

511493-- from LAoP.

512494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

513−comp Empty Empty = Empty

514−comp (One a) (One b) = One (a * b)

515−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law

516−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law

517−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law

495+comp (One a) (One b) = One (a * b)

496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law

497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law

498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law

518499{-# NOINLINE comp #-}

519−{-# RULES

520− "comp/identity1" forall m. comp m identity = m ;

521− "comp/identity2" forall m. comp identity m = m

500+{-# RULES

501+ "comp/iden1" forall m. comp m iden = m ;

502+ "comp/iden2" forall m. comp iden m = m

522503#-}

523504

505+-- Scalar multiplication of matrices

506+

507+infixl 7 .|

508+-- | Scalar multiplication of matrices.

509+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

510+(.|) e (One a) = One (e * a)

511+(.|) e (Join a b) = Join (e .| a) (e .| b)

512+(.|) e (Fork a b) = Fork (e .| a) (e .| b)

513+

514+-- Scalar division of matrices

515+

516+infixl 7 ./

517+-- | Scalar multiplication of matrices.

518+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

519+(./) (One a) e = One (a / e)

520+(./) (Join a b) e = Join (a ./ e) (b ./ e)

521+(./) (Fork a b) e = Fork (a ./ e) (b ./ e)

522+

524523-- Projections

525524

526525-- | Biproduct first component projection

527−p1 :: forall e m n. (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e (Either m n) m

528−p1 =

529− let iden = identity :: Matrix e m m

530− zero = zeros :: Matrix e n m

531− in junc iden zero

526+p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m

527+p1 = join id zeros

532528

533529-- | Biproduct second component projection

534−p2 :: forall e m n. (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e (Either m n) n

535−p2 =

536− let iden = identity :: Matrix e n n

537− zero = zeros :: Matrix e m n

538− in junc zero iden

530+p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n

531+p2 = join zeros id

539532

540533-- Injections

541534

542535-- | Biproduct first component injection

543−i1 :: (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e m (Either m n)

536+i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)

544537i1 = tr p1

545538

546539-- | Biproduct second component injection

547−i2 :: (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e n (Either m n)

540+i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)

548541i2 = tr p2

549542

550543-- Dimensions

551544

552545-- | Obtain the number of rows.

553546--

554−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

555−-- dimensions in constant time.

556−--

557−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

558−-- exchange for performance.

547+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

548+-- dimensions in constant time. For a version that doesn't require the

549+-- constraint see 'rows''.

559550rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int

560551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))

561552

553+-- | Obtain the number of rows in an inefficient manner, but without any

554+-- constraints.

555+--

556+-- For a more efficient version see 'rows'.

557+rows' :: Matrix e cols rows -> Int

558+rows' (One _) = 1

559+rows' (Join lhs _) = rows' lhs

560+rows' (Fork top bottom) = rows' top + rows' bottom

561+

562562-- | Obtain the number of columns.

563−--

564−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

565−-- dimensions in constant time.

566563--

567−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

568−-- exchange for performance.

564+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

565+-- dimensions in constant time. For a version that doesn't require the

566+-- constraint see 'columns''.

569567columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int

570568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))

571569

570+-- | Obtain the number of columns in an inefficient manner, but without any

571+-- constraints.

572+--

573+-- For a more efficient version see 'columns'.

574+columns' :: Matrix e cols rows -> Int

575+columns' (One _) = 1

576+columns' (Join lhs rhs) = columns' lhs + columns' rhs

577+columns' (Fork top _) = columns' top

578+

572579-- Coproduct Bifunctor

573580

574581infixl 5 -|-

@@ -577,164 +584,161 @@ infixl 5 -|-

577584(-|-) ::

578585 forall e n k m j.

579586 ( Num e,

580− CountableDimensions j k,

581− FromLists e k k,

582− FromLists e j k,

583− FromLists e k j,

584− FromLists e j j

587+ CountableDims j k,

588+ FL k k,

589+ FL j k,

590+ FL k j,

591+ FL j j

585592 ) =>

586593 Matrix e n k ->

587594 Matrix e m j ->

588595 Matrix e (Either n m) (Either k j)

589−(-|-) a b = Junc (i1 . a) (i2 . b)

596+(-|-) a b = Join (i1 . a) (i2 . b)

590597

591598-- Khatri Rao Product and projections

592599

593600-- | Khatri Rao product first component projection matrix.

594−kp1 ::

601+fstM ::

595602 forall e m k .

596603 ( Num e,

597− CountableDimensions k m,

598− FromLists e (Normalize (m, k)) m,

604+ CountableDims k m,

605+ FL (Normalize (m, k)) m,

599606 CountableN (m, k)

600607 ) => Matrix e (Normalize (m, k)) m

601−kp1 = matrixBuilder f

608+fstM = matrixBuilder' f

602609 where

603610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

604611 f (x, y)

605612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1

606− | otherwise = 0

613+ | otherwise = 0

607614

608615-- | Khatri Rao product second component projection matrix.

609−kp2 ::

616+sndM ::

610617 forall e m k .

611618 ( Num e,

612− CountableDimensions k m,

613− FromLists e (Normalize (m, k)) k,

619+ CountableDims k m,

620+ FL (Normalize (m, k)) k,

614621 CountableN (m, k)

615622 ) => Matrix e (Normalize (m, k)) k

616−kp2 = matrixBuilder f

623+sndM = matrixBuilder' f

617624 where

618625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

619626 f (x, y)

620627 | x == y || mod (y - x) offset == 0 = 1

621− | otherwise = 0

628+ | otherwise = 0

622629

623630-- | Khatri Rao Matrix product also known as matrix pairing.

624631--

625632-- NOTE: That this is not a true categorical product, see for instance:

626−--

633+--

627634-- @

628−-- | kp1 . khatri a b == a

629−-- khatri a b ==> |

630−-- | kp2 . khatri a b == b

635+-- | fstM . kr a b == a

636+-- kr a b ==> |

637+-- | sndM . kr a b == b

631638-- @

632639--

633640-- __Emphasis__ on the implication symbol.

634−khatri ::

635− forall e cols a b.

641+kr ::

642+ forall e cols a b.

636643 ( Num e,

637− CountableDimensions a b,

644+ CountableDims a b,

638645 CountableN (a, b),

639− FromLists e (Normalize (a, b)) a,

640− FromLists e (Normalize (a, b)) b

646+ FL (Normalize (a, b)) a,

647+ FL (Normalize (a, b)) b

641648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))

642−khatri a b =

643− let kp1' = kp1 @e @a @b

644− kp2' = kp2 @e @a @b

645− in (tr kp1') . a * (tr kp2') . b

649+kr a b =

650+ let fstM' = fstM @e @a @b

651+ sndM' = sndM @e @a @b

652+ in (tr fstM' . a) * (tr sndM' . b)

646653

647654-- Product Bifunctor (Kronecker)

648655

649656infixl 4 ><

650657

651658-- | Matrix product functor also known as kronecker product

652−(><) ::

653− forall e m p n q.

659+(><) ::

660+ forall e m p n q.

654661 ( Num e,

655− CountableDimensions m n,

656− CountableDimensions p q,

657− CountableDimensionsN (m, n) (p, q),

658− FromLists e (Normalize (m, n)) m,

659− FromLists e (Normalize (m, n)) n,

660− FromLists e (Normalize (p, q)) p,

661− FromLists e (Normalize (p, q)) q

662− )

662+ CountableDims m n,

663+ CountableDims p q,

664+ CountableDimsN (m, n) (p, q),

665+ FL (Normalize (m, n)) m,

666+ FL (Normalize (m, n)) n,

667+ FL (Normalize (p, q)) p,

668+ FL (Normalize (p, q)) q

669+ )

663670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))

664671(><) a b =

665− let kp1' = kp1 @e @m @n

666− kp2' = kp2 @e @m @n

667− in khatri (a . kp1') (b . kp2')

672+ let fstM' = fstM @e @m @n

673+ sndM' = sndM @e @m @n

674+ in kr (a . fstM') (b . sndM')

668675

669−-- Matrix abide Junc Split

676+-- Matrix abide Join Fork

670677

671−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

672−--

678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.

679+--

673680-- Law:

674681--

675682-- @

676−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

683+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

677684-- @

678−abideJS :: Matrix e cols rows -> Matrix e cols rows

679−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law

680−abideJS Empty = Empty

681−abideJS (One e) = One e

682−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)

683−abideJS (Split a b) = Split (abideJS a) (abideJS b)

685+abideJF :: Matrix e cols rows -> Matrix e cols rows

686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law

687+abideJF (One e) = One e

688+abideJF (Join a b) = Join (abideJF a) (abideJF b)

689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)

684690

685−-- Matrix abide Split Junc

691+-- Matrix abide Fork Join

686692

687−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

688−--

693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

694+--

689695-- @

690−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

696+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

691697-- @

692−abideSJ :: Matrix e cols rows -> Matrix e cols rows

693−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law

694−abideSJ Empty = Empty

695−abideSJ (One e) = One e

696−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)

697−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)

698+abideFJ :: Matrix e cols rows -> Matrix e cols rows

699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law

700+abideFJ (One e) = One e

701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)

702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)

698703

699704-- Matrix transposition

700705

701706-- | Matrix transposition.

702707tr :: Matrix e cols rows -> Matrix e rows cols

703−tr Empty = Empty

704−tr (One e) = One e

705−tr (Junc a b) = Split (tr a) (tr b)

706−tr (Split a b) = Junc (tr a) (tr b)

708+tr (One e) = One e

709+tr (Join a b) = Fork (tr a) (tr b)

710+tr (Fork a b) = Join (tr a) (tr b)

707711

708712-- Selective 'select' operator

709713

710714-- | Selective functors 'select' operator equivalent inspired by the

711715-- ArrowMonad solution presented in the paper.

712−select :: (Num e, FromLists e b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

713−select (Split a b) y = y . a + b -- Divide-and-conquer law

714−select (Junc (Split a c) (Split b d)) y = junc (y . a + c) (y . b + d) -- Pattern matching + DnC law

715−select m y = junc y identity . m

716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

717+select (Fork a b) y = y . a + b -- Divide-and-conquer law

718+select (Join (Fork a c) (Fork b d)) y = join (y . a + c) (y . b + d) -- Pattern matching + DnC law

719+select m y = join y id . m

716720

717721branch ::

718722 ( Num e,

719− CountableDimensions a b,

720− CountableDimensions c (Either b c),

721− FromLists e c b,

722− FromLists e a b,

723− FromLists e a a,

724− FromLists e b b,

725− FromLists e c c,

726− FromLists e b a,

727− FromLists e b c,

728− FromLists e (Either b c) b,

729− FromLists e (Either b c) c

723+ CountableDims a b,

724+ CountableDims c (Either b c),

725+ FL c b,

726+ FL a b,

727+ FL a a,

728+ FL b b,

729+ FL c c,

730+ FL b a,

731+ FL b c,

732+ FL (Either b c) b,

733+ FL (Either b c) c

730734 )

731735 => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c

732736branch x l r = f x `select` g l `select` r

733737 where

734− f :: (Num e, Countable a, CountableDimensions b c, FromLists e a b, FromLists e c b, FromLists e b b, FromLists e b a, FromLists e a a)

738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)

735739 => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))

736− f m = split (tr i1) (i1 . tr i2) . m

737− g :: (Num e, CountableDimensions b c, FromLists e b c, FromLists e c c) => Matrix e a c -> Matrix e a (Either b c)

740+ f m = fork (tr i1) (i1 . tr i2) . m

741+ g :: (Num e, CountableDims b c, FL b c, FL c c) => Matrix e a c -> Matrix e a (Either b c)

738742 g m = i2 . m

739743

740744-- McCarthy's Conditional

@@ -743,9 +747,9 @@ branch x l r = f x `select` g l `select` r

743747cond ::

744748 ( Trivial cols,

745749 Countable cols,

746− FromLists e () cols,

747− FromLists e cols (),

748− FromLists e cols cols,

750+ FL () cols,

751+ FL cols (),

752+ FL cols cols,

749753 Bounded a,

750754 Enum a,

751755 Num e,

@@ -753,14 +757,14 @@ cond ::

753757 )

754758 =>

755759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

756−cond p f g = junc f g . grd p

760+cond p f g = join f g . grd p

757761

758−grd ::

762+grd ::

759763 ( Trivial q,

760764 Countable q,

761− FromLists e () q,

762− FromLists e q (),

763− FromLists e q q,

765+ FL () q,

766+ FL q (),

767+ FL q q,

764768 Bounded a,

765769 Enum a,

766770 Num e,

@@ -768,44 +772,44 @@ grd ::

768772 )

769773 =>

770774 (a -> Bool) -> Matrix e q (Either q q)

771−grd f = split (corr f) (corr (not . f))

775+grd f = fork (corr f) (corr (not . f))

772776

773−corr ::

774− forall e a q .

777+corr ::

778+ forall e a q .

775779 ( Trivial q,

776780 Countable q,

777− FromLists e () q,

778− FromLists e q (),

779− FromLists e q q,

781+ FL () q,

782+ FL q (),

783+ FL q q,

780784 Liftable e a Bool

781− )

785+ )

782786 => (a -> Bool) -> Matrix e q q

783−corr p = let f = fromF p :: Matrix e q ()

784− in khatri f (identity :: Matrix e q q)

787+corr p = let f = fromF' p :: Matrix e q ()

788+ in kr f (id :: Matrix e q q)

785789

786790-- Pretty print

787791

788792prettyAux :: Show e => [[e]] -> [[e]] -> String

789793prettyAux [] _ = ""

790−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

791795 where

792796 v = fmap show m

793797 widest = maximum $ fmap length v

794798 fill str = replicate (widest - length str - 2) ' ' ++ str

795−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

796800 where

797− v = fmap show m

798− widest = maximum $ fmap length v

801+ v = fmap show m

802+ widest = maximum $ fmap length v

799803 fill str = replicate (widest - length str - 2) ' ' ++ str

800−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

801805 prettyAux t l

802806 where

803− v = fmap show l

804− widest = maximum $ fmap length v

807+ v = fmap show l

808+ widest = maximum $ fmap length v

805809 fill str = replicate (widest - length str - 2) ' ' ++ str

806810

807811-- | Matrix pretty printer

808−pretty :: (CountableDimensions cols rows, Show e) => Matrix e cols rows -> String

812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String

809813pretty m = concat

810814 [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"

811815 , unlines

@@ -813,16 +817,16 @@ pretty m = concat

813817 , "└ ", unwords (replicate (columns m) blank), " ┘"

814818 ]

815819 where

816− strings = map show (toList m)

817− widest = maximum $ map length strings

820+ strings = map show (toList m)

821+ widest = maximum $ map length strings

818822 fill str = replicate (widest - length str) ' ' ++ str

819− blank = fill ""

823+ blank = fill ""

820824 safeGet i j m

821825 | i > rows m || j > columns m || i < 1 || j < 1 = Nothing

822826 | otherwise = Just $ unsafeGet i j m (toList m)

823827 unsafeGet i j m l = l !! encode (columns m) (i,j)

824− encode m (i,j) = (i-1)*m + j - 1

825− getElem i j m =

828+ encode m (i,j) = (i-1)*m + j - 1

829+ getElem i j m =

826830 fromMaybe

827831 (error $

828832 "getElem: Trying to get the "

@@ -834,12 +838,20 @@ pretty m = concat

834838 (safeGet i j m)

835839

836840-- | Matrix pretty printer

837−prettyPrint :: (CountableDimensions cols rows, Show e) => Matrix e cols rows -> IO ()

841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()

838842prettyPrint = putStrLn . pretty

839843

844+-- | Zip two matrices with a given binary function

845+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows

846+zipWithM f (One a) (One b) = One (f a b)

847+zipWithM f (Join a b) (Join c d) = Join (zipWithM f a c) (zipWithM f b d)

848+zipWithM f (Fork a b) (Fork c d) = Fork (zipWithM f a c) (zipWithM f b d)

849+zipWithM f x@(Fork _ _) y@(Join _ _) = zipWithM f x (abideJF y)

850+zipWithM f x@(Join _ _) y@(Fork _ _) = zipWithM f (abideJF x) y

851+

840852-- Relational operators functions

841853

842−type Boolean = Natural 0 1

854+type Boolean = Natural 0 1

843855type Relation a b = Matrix Boolean a b

844856

845857-- | Helper conversion function

@@ -855,54 +867,47 @@ fromBool False = nat 0

855867

856868-- | Relational negation

857869negateM :: Relation cols rows -> Relation cols rows

858−negateM Empty = Empty

859−negateM (One (Nat 0)) = One (Nat 1)

860−negateM (One (Nat 1)) = One (Nat 0)

861−negateM (Junc a b) = Junc (negateM a) (negateM b)

862−negateM (Split a b) = Split (negateM a) (negateM b)

870+negateM (One (Nat p)) = One (Nat (negate p))

871+negateM (Join a b) = Join (negateM a) (negateM b)

872+negateM (Fork a b) = Fork (negateM a) (negateM b)

863873

864874-- | Relational addition

865875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows

866−orM Empty Empty = Empty

867−orM (One a) (One b) = One (fromBool (toBool a || toBool b))

868−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)

869−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)

870−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)

871−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y

876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))

877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)

878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)

879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)

880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y

872881

873882-- | Relational multiplication

874883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows

875−andM Empty Empty = Empty

876−andM (One a) (One b) = One (fromBool (toBool a && toBool b))

877−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)

878−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)

879−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)

880−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y

884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))

885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)

886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)

887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)

888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y

881889

882890-- | Relational subtraction

883891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows

884−subM Empty Empty = Empty

885−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

886−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)

887−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)

888−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)

889−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y

892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)

894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)

895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)

896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y

890897

891898-- | Matrix relational composition.

892899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows

893−compRel Empty Empty = Empty

894−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

895−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

896−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law

897−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law

900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law

903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law

898904

899905-- | Matrix relational right division

900906divR :: Relation b c -> Relation b a -> Relation a c

901−divR Empty Empty = Empty

902907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a

903−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)

904−divR (Split a b) c = Split (divR a c) (divR b c)

905−divR c (Split a b) = Junc (divR c a) (divR c b)

908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)

909+divR (Fork a b) c = Fork (divR a c) (divR b c)

910+divR c (Fork a b) = Join (divR c a) (divR c b)

906911

907912-- | Matrix relational left division

908913divL :: Relation c b -> Relation a b -> Relation a c

@@ -918,86 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)

918923--

919924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of

920925-- types @a@ or @b@ allows.

921−fromFRel ::

926+fromFRel' ::

922927 forall a b cols rows.

923928 ( Liftable Boolean a b,

924− CountableDimensions cols rows,

925− FromLists Boolean rows cols

929+ CountableDims cols rows,

930+ FL rows cols

926931 ) =>

927932 (a -> b) ->

928933 Relation cols rows

929−fromFRel f =

930− let minA = minBound @a

931− maxA = maxBound @a

932− minB = minBound @b

933− maxB = maxBound @b

934− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))

935− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))

936− elementsA = take ccols [minA .. maxA]

937− elementsB = take rrows [minB .. maxB]

938− combinations = (,) <$> elementsA <*> elementsB

939− combAp = map snd . sort . map (\(a, b) -> if f a == b

940− then ((fromEnum a, fromEnum b), nat 1)

941− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

942− mList = buildList combAp rrows

943− in tr $ fromLists mList

944− where

945− buildList [] _ = []

946− buildList l r = take r l : buildList (drop r l) r

934+fromFRel' = fromF'

947935

948936-- | Lifts functions to relations with dimensions matching @a@ and @b@

949937-- cardinality's.

950−fromFRel' ::

938+fromFRel ::

951939 forall a b.

952940 ( Liftable Boolean a b,

953− CountableDimensionsN a b,

954− FromLists Boolean (Normalize b) (Normalize a)

941+ CountableDimsN a b,

942+ FLN b a

955943 ) =>

956944 (a -> b) ->

957945 Relation (Normalize a) (Normalize b)

958−fromFRel' f =

959− let minA = minBound @a

960− maxA = maxBound @a

961− minB = minBound @b

962− maxB = maxBound @b

963− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

964− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

965− elementsA = take ccols [minA .. maxA]

966− elementsB = take rrows [minB .. maxB]

967− combinations = (,) <$> elementsA <*> elementsB

968− combAp = map snd . sort . map (\(a, b) -> if f a == b

969− then ((fromEnum a, fromEnum b), nat 1)

970− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

971− mList = buildList combAp rrows

972− in tr $ fromLists mList

973− where

974− buildList [] _ = []

975− buildList l r = take r l : buildList (drop r l) r

946+fromFRel = fromFRel'

976947

977948-- | Lifts a relation function to a Boolean Matrix

978−toRel ::

949+toRel ::

979950 forall a b.

980951 ( Bounded a,

981952 Bounded b,

982953 Enum a,

983954 Enum b,

984955 Eq b,

985− CountableDimensionsN a b,

986− FromListsN Boolean b a

987− )

956+ CountableDims a b,

957+ FLN b a

958+ )

988959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)

989960toRel f =

990961 let minA = minBound @a

991962 maxA = maxBound @a

992963 minB = minBound @b

993964 maxB = maxBound @b

994− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

995− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))

966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))

996967 elementsA = take ccols [minA .. maxA]

997968 elementsB = take rrows [minB .. maxB]

998969 combinations = (,) <$> elementsA <*> elementsB

999− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

1000− then ((fromEnum a, fromEnum b), nat 1)

970+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

971+ then ((fromEnum a, fromEnum b), nat 1)

1001972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations

1002973 mList = buildList combAp rrows

1003974 in tr $ fromLists mList

src/LAoP/Matrix/Nat.hs+139−124

@@ -56,21 +56,20 @@ module LAoP.Matrix.Nat

5656

5757 -- * Constraint type synonyms

5858 Countable,

59− CountableDimensions,

59+ CountableDims,

6060 CountableN,

6161 CountableNz,

62− CountableDimensionsN,

63− FromListsN,

64− FromListsNz,

62+ CountableDimsN,

63+ FLN,

64+ FLNz,

6565 Liftable,

6666 TrivialE,

6767 TrivialP,

6868

6969 -- * Primitives

70− empty,

7170 one,

72− junc,

73− split,

71+ join,

72+ fork,

7473

7574 -- * Auxiliary type families

7675 I.FromNat,

@@ -78,11 +77,11 @@ module LAoP.Matrix.Nat

7877 I.Normalize,

7978

8079 -- * Matrix construction and conversion

81− I.FromLists,

80+ I.FL,

8281 fromLists,

8382 toLists,

8483 toList,

85− matrixBuilder,

84+ matrixBuilder',

8685 row,

8786 col,

8887 zeros,

@@ -98,6 +97,10 @@ module LAoP.Matrix.Nat

9897 -- ** Matrix Transposition

9998 tr,

10099

100+ -- ** Scalar multiplication/division of matrices

101+ (.|),

102+ (./),

103+

101104 -- ** Selective operator

102105 select,

103106

@@ -105,16 +108,19 @@ module LAoP.Matrix.Nat

105108 cond,

106109

107110 -- ** Matrix "abiding"

108− abideJS,

109− abideSJ,

111+ abideJF,

112+ abideFJ,

113+

114+ -- ** Zip Matrices

115+ zipWithM,

110116

111117 -- * Biproduct approach

112− -- ** Split

118+ -- ** Fork

113119 (===),

114120 -- *** Projections

115121 p1,

116122 p2,

117− -- ** Junc

123+ -- ** Join

118124 (|||),

119125 -- *** Injections

120126 i1,

@@ -128,11 +134,11 @@ module LAoP.Matrix.Nat

128134 -- | Note that given the restrictions imposed it is not possible to

129135 -- implement the standard type classes present in standard Haskell.

130136 -- *** Matrix pairing projections

131− kp1,

132− kp2,

137+ fstM,

138+ sndM,

133139

134140 -- *** Matrix pairing

135− khatri,

141+ kr,

136142

137143 -- * Matrix composition and lifting

138144

@@ -140,10 +146,10 @@ module LAoP.Matrix.Nat

140146

141147 -- | Note that given the restrictions imposed it is not possible to

142148 -- implement the standard type classes present in standard Haskell.

143− identity,

149+ iden,

144150 comp,

145− fromF,

146151 fromF',

152+ fromF,

147153

148154 -- * Matrix printing

149155 pretty,

@@ -154,47 +160,34 @@ where

154160import Data.Proxy

155161import GHC.TypeLits

156162import Control.DeepSeq

157−import qualified Control.Category as C

158163import qualified LAoP.Matrix.Internal as I

159164

160165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))

161166 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))

162167

163168-- | Constraint type synonyms to keep the type signatures less convoluted

164−type Countable a = KnownNat (I.Count a)

165−type CountableDimensions a b = (Countable a, Countable b)

166−type CountableN a = KnownNat (I.Count (I.FromNat a))

167−type CountableNz a = KnownNat (I.Count (I.Normalize a))

168−type CountableDimensionsN a b = (CountableN a, CountableN b)

169−type FromListsN e a b = I.FromLists e (I.FromNat a) (I.FromNat b)

170−type FromListsNz e a b = I.FromLists e (I.Normalize a) (I.Normalize b)

171−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

172−type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)

173−type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))

174−

175−-- | It isn't possible to implement the 'id' function so it's

176−-- implementation is 'undefined'. However 'comp' can be and this partial

177−-- class implementation exists just to make the code more readable.

178−--

179−-- Please use 'identity' instead.

180−instance (Num e) => C.Category (Matrix e) where

181− id = undefined

182− (.) = comp

169+type Countable a = KnownNat (I.Count a)

170+type CountableDims a b = (Countable a, Countable b)

171+type CountableN a = KnownNat (I.Count (I.FromNat a))

172+type CountableNz a = KnownNat (I.Count (I.Normalize a))

173+type CountableDimsN a b = (CountableN a, CountableN b)

174+type FLN e a b = I.FL (I.FromNat a) (I.FromNat b)

175+type FLNz e a b = I.FL (I.Normalize a) (I.Normalize b)

176+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

177+type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)

178+type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))

183179

184180-- Primitives

185181

186−empty :: Matrix e 0 0

187−empty = M I.Empty

188−

189182one :: e -> Matrix e 1 1

190183one = M . I.One

191184

192−junc ::

185+join ::

193186 (TrivialE a b) =>

194187 Matrix e a rows ->

195188 Matrix e b rows ->

196189 Matrix e (a + b) rows

197−junc (M a) (M b) = M (I.Junc a b)

190+join (M a) (M b) = M (I.Join a b)

198191

199192infixl 3 |||

200193(|||) ::

@@ -202,14 +195,14 @@ infixl 3 |||

202195 Matrix e a rows ->

203196 Matrix e b rows ->

204197 Matrix e (a + b) rows

205−(|||) = junc

198+(|||) = join

206199

207−split ::

200+fork ::

208201 (TrivialE a b) =>

209202 Matrix e cols a ->

210203 Matrix e cols b ->

211204 Matrix e cols (a + b)

212−split (M a) (M b) = M (I.Split a b)

205+fork (M a) (M b) = M (I.Fork a b)

213206

214207infixl 2 ===

215208(===) ::

@@ -217,43 +210,45 @@ infixl 2 ===

217210 Matrix e cols a ->

218211 Matrix e cols b ->

219212 Matrix e cols (a + b)

220−(===) = split

213+(===) = fork

221214

222215-- Construction

223216

224−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows

217+fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows

225218fromLists = M . I.fromLists

226219

227−matrixBuilder ::

228− (FromListsN e cols rows, CountableN cols, CountableN rows)

220+-- | Matrix builder function. Constructs a matrix provided with

221+-- a construction function that operates with indices.

222+matrixBuilder' ::

223+ (FLN e cols rows, CountableN cols, CountableN rows)

229224 => ((Int, Int) -> e) -> Matrix e cols rows

230−matrixBuilder = M . I.matrixBuilder

225+matrixBuilder' = M . I.matrixBuilder'

231226

232−col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows

227+col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows

233228col = M . I.col

234229

235−row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1

230+row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1

236231row = M . I.row

237232

238−fromF ::

233+fromF' ::

239234 ( Liftable e a b,

240235 CountableN cols,

241236 CountableN rows,

242− FromListsN e rows cols

237+ FLN e rows cols

243238 ) =>

244239 (a -> b) ->

245240 Matrix e cols rows

246−fromF = M . I.fromF

241+fromF' = M . I.fromF'

247242

248−fromF' ::

243+fromF ::

249244 ( Liftable e a b,

250245 CountableNz a,

251246 CountableNz b,

252− FromListsNz e b a

247+ FLNz e b a

253248 ) =>

254249 (a -> b) ->

255250 Matrix e (I.Count a) (I.Count b)

256−fromF' = undefined -- M . I.fromF'

251+fromF = undefined -- M . I.fromF

257252

258253-- Conversion

259254

@@ -266,21 +261,21 @@ toList (M m) = I.toList m

266261-- Zeros Matrix

267262

268263zeros ::

269− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

264+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

270265 Matrix e cols rows

271266zeros = M I.zeros

272267

273268-- Ones Matrix

274269

275270ones ::

276− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

271+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

277272 Matrix e cols rows

278273ones = M I.ones

279274

280275-- Const Matrix

281276

282277constant ::

283− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

278+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

284279 e ->

285280 Matrix e cols rows

286281constant = M . I.constant

@@ -289,27 +284,41 @@ constant = M . I.constant

289284

290285bang ::

291286 forall e cols.

292− (Num e, Enum e, I.FromLists e (I.FromNat cols) (), CountableN cols) =>

287+ (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>

293288 Matrix e cols 1

294289bang = M I.bang

295290

296−-- Identity Matrix

291+-- iden Matrix

297292

298−identity ::

299− (Num e, FromListsN e cols cols, CountableN cols) =>

293+iden ::

294+ (Num e, FLN e cols cols, CountableN cols) =>

300295 Matrix e cols cols

301−identity = M I.identity

296+iden = M I.iden

302297

303298-- Matrix composition (MMM)

304299

305300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

306301comp (M a) (M b) = M (I.comp a b)

307302

303+-- Scalar multiplication of matrices

304+

305+infixl 7 .|

306+-- | Scalar multiplication of matrices.

307+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

308+(.|) e (M m) = M (e I..| m)

309+

310+-- Scalar division of matrices

311+

312+infixl 7 ./

313+-- | Scalar multiplication of matrices.

314+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

315+(./) (M m) e = M (m I../ e)

316+

308317p1 ::

309318 ( Num e,

310− CountableDimensionsN n m,

311− FromListsN e n m,

312− FromListsN e m m,

319+ CountableDimsN n m,

320+ FLN e n m,

321+ FLN e m m,

313322 TrivialE m n

314323 ) =>

315324 Matrix e (m + n) m

@@ -317,9 +326,9 @@ p1 = M I.p1

317326

318327p2 ::

319328 ( Num e,

320− CountableDimensionsN n m,

321− FromListsN e m n,

322− FromListsN e n n,

329+ CountableDimsN n m,

330+ FLN e m n,

331+ FLN e n n,

323332 TrivialE m n

324333 ) =>

325334 Matrix e (m + n) n

@@ -329,9 +338,9 @@ p2 = M I.p2

329338

330339i1 ::

331340 ( Num e,

332− CountableDimensionsN n rows,

333− FromListsN e n rows,

334− FromListsN e rows rows,

341+ CountableDimsN n rows,

342+ FLN e n rows,

343+ FLN e rows rows,

335344 TrivialE rows n

336345 ) =>

337346 Matrix e rows (rows + n)

@@ -339,9 +348,9 @@ i1 = tr p1

339348

340349i2 ::

341350 ( Num e,

342− CountableDimensionsN rows m,

343− FromListsN e m rows,

344− FromListsN e rows rows,

351+ CountableDimsN rows m,

352+ FLN e m rows,

353+ FLN e rows rows,

345354 TrivialE m rows

346355 ) =>

347356 Matrix e rows (m + rows)

@@ -355,16 +364,16 @@ rows (M m) = I.rows m

355364columns :: (CountableN cols) => Matrix e cols rows -> Int

356365columns (M m) = I.columns m

357366

358−-- | Coproduct Bifunctor

359367infixl 5 -|-

360368

369+-- | Coproduct Bifunctor (Direct sum)

361370(-|-) ::

362371 ( Num e,

363− CountableDimensionsN j k,

364− FromListsN e k k,

365− FromListsN e j k,

366− FromListsN e k j,

367− FromListsN e j j,

372+ CountableDimsN j k,

373+ FLN e k k,

374+ FLN e j k,

375+ FLN e k j,

376+ FLN e j j,

368377 TrivialE n m,

369378 TrivialE k j

370379 ) =>

@@ -373,40 +382,42 @@ infixl 5 -|-

373382 Matrix e (n + m) (k + j)

374383(-|-) (M a) (M b) = M ((I.-|-) a b)

375384

376−-- | Khatri Rao Product and projections

377−kp1 ::

385+-- | Khatri Rao Product first projection

386+fstM ::

378387 forall e m k .

379388 ( Num e,

380− CountableDimensionsN m k,

389+ CountableDimsN m k,

381390 CountableN (m * k),

382− FromListsN e (m * k) m,

391+ FLN e (m * k) m,

383392 TrivialP m k

384393 ) => Matrix e (m * k) m

385−kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))

394+fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))

386395

387−kp2 ::

396+-- | Khatri Rao Product second projection

397+sndM ::

388398 forall e m k.

389399 ( Num e,

390− CountableDimensionsN k m,

391− FromListsN e (m * k) k,

400+ CountableDimsN k m,

401+ FLN e (m * k) k,

392402 CountableN (m * k),

393403 TrivialP m k

394404 ) => Matrix e (m * k) k

395−kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))

405+sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))

396406

397−khatri ::

407+-- | Khatri Rao Product

408+kr ::

398409 forall e cols a b.

399410 ( Num e,

400− CountableDimensionsN a b,

411+ CountableDimsN a b,

401412 CountableN (a * b),

402− FromListsN e (a * b) a,

403− FromListsN e (a * b) b,

413+ FLN e (a * b) a,

414+ FLN e (a * b) b,

404415 TrivialP a b

405416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)

406−khatri a b =

407− let kp1' = kp1 @e @a @b

408− kp2' = kp2 @e @a @b

409− in comp (tr kp1') a * comp (tr kp2') b

417+kr a b =

418+ let fstM' = fstM @e @a @b

419+ sndM' = sndM @e @a @b

420+ in comp (tr fstM') a * comp (tr sndM') b

410421

411422-- | Product Bifunctor (Kronecker)

412423infixl 4 ><

@@ -414,28 +425,28 @@ infixl 4 ><

414425(><) ::

415426 forall e m p n q.

416427 ( Num e,

417− CountableDimensionsN m n,

418− CountableDimensionsN p q,

419− CountableDimensionsN (m * n) (p * q),

420− FromListsN e (m * n) m,

421− FromListsN e (m * n) n,

422− FromListsN e (p * q) p,

423− FromListsN e (p * q) q,

428+ CountableDimsN m n,

429+ CountableDimsN p q,

430+ CountableDimsN (m * n) (p * q),

431+ FLN e (m * n) m,

432+ FLN e (m * n) n,

433+ FLN e (p * q) p,

434+ FLN e (p * q) q,

424435 TrivialP m n,

425436 TrivialP p q

426437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)

427438(><) a b =

428− let kp1' = kp1 @e @m @n

429− kp2' = kp2 @e @m @n

430− in khatri (comp a kp1') (comp b kp2')

439+ let fstM' = fstM @e @m @n

440+ sndM' = sndM @e @m @n

441+ in kr (comp a fstM') (comp b sndM')

431442

432−-- | Matrix abide Junc Split

433−abideJS :: Matrix e cols rows -> Matrix e cols rows

434−abideJS (M m) = M (I.abideJS m)

443+-- | Matrix abide Join Fork

444+abideJF :: Matrix e cols rows -> Matrix e cols rows

445+abideJF (M m) = M (I.abideJF m)

435446

436−-- | Matrix abide Split Junc

437−abideSJ :: Matrix e cols rows -> Matrix e cols rows

438−abideSJ (M m) = M (I.abideSJ m)

447+-- | Matrix abide Fork Join

448+abideFJ :: Matrix e cols rows -> Matrix e cols rows

449+abideFJ (M m) = M (I.abideFJ m)

439450

440451-- | Matrix transposition

441452tr :: Matrix e cols rows -> Matrix e rows cols

@@ -444,7 +455,7 @@ tr (M m) = M (I.tr m)

444455-- Selective 'select' operator

445456select ::

446457 ( Num e,

447− FromListsN e rows1 rows1,

458+ FLN e rows1 rows1,

448459 CountableN rows1,

449460 I.FromNat rows2 ~ I.FromNat rows1,

450461 I.FromNat cols1 ~ I.FromNat cols2,

@@ -457,9 +468,9 @@ select (M m) (M y) = M (I.select m y)

457468cond ::

458469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,

459470 CountableN cols,

460− I.FromLists e () (I.FromNat cols),

461− I.FromLists e (I.FromNat cols) (),

462− FromListsN e cols cols,

471+ I.FL () (I.FromNat cols),

472+ I.FL (I.FromNat cols) (),

473+ FLN e cols cols,

463474 Liftable e a Bool

464475 )

465476 =>

@@ -468,8 +479,12 @@ cond p (M a) (M b) = M (I.cond p a b)

468479

469480-- Pretty print

470481

471−pretty :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> String

482+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

472483pretty (M m) = I.pretty m

473484

474−prettyPrint :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> IO ()

485+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

475486prettyPrint (M m) = I.prettyPrint m

487+

488+-- | Zip two matrices with a given binary function

489+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows

490+zipWithM f (M a) (M b) = M (I.zipWithM f a b)

src/LAoP/Matrix/Type.hs+231−175

@@ -1,11 +1,11 @@

11{-# LANGUAGE ConstraintKinds #-}

2+{-# LANGUAGE TypeFamilies #-}

23{-# LANGUAGE AllowAmbiguousTypes #-}

34{-# LANGUAGE DataKinds #-}

45{-# LANGUAGE DerivingVia #-}

56{-# LANGUAGE FlexibleContexts #-}

67{-# LANGUAGE FlexibleInstances #-}

78{-# LANGUAGE GADTs #-}

8−{-# LANGUAGE KindSignatures #-}

99{-# LANGUAGE MultiParamTypeClasses #-}

1010{-# LANGUAGE NoStarIsType #-}

1111{-# LANGUAGE RankNTypes #-}

@@ -58,10 +58,10 @@ module LAoP.Matrix.Type

5858

5959 -- * Constraint type aliases

6060 Countable,

61− CountableDimensions,

61+ CountableDims,

6262 CountableN,

63− CountableDimensionsN,

64− FromListsN,

63+ CountableDimsN,

64+ FLN,

6565 Liftable,

6666 Trivial,

6767 TrivialP,

@@ -71,10 +71,9 @@ module LAoP.Matrix.Type

7171 One,

7272

7373 -- * Primitives

74− empty,

7574 one,

76− junc,

77− split,

75+ join,

76+ fork,

7877

7978 -- * Auxiliary type families

8079 I.FromNat,

@@ -86,6 +85,7 @@ module LAoP.Matrix.Type

8685 fromLists,

8786 toLists,

8887 toList,

88+ matrixBuilder',

8989 matrixBuilder,

9090 row,

9191 col,

@@ -97,14 +97,15 @@ module LAoP.Matrix.Type

9797

9898 -- * Functor instance equivalent function

9999 fmapM,

100+ bimapM,

100101

101102 -- * Applicative/Monoidal instance equivalent functions

102103 unitM,

103104 multM,

104105

105106 -- * Selective equivalent instance function

106− selectM,

107−

107+ selectM,

108+

108109 -- * Monad equivalent instance function

109110 returnM,

110111 bindM,

@@ -112,25 +113,34 @@ module LAoP.Matrix.Type

112113 -- * Misc

113114 -- ** Get dimensions

114115 columns,

116+ columns',

115117 rows,

118+ rows',

116119

117120 -- ** Matrix Transposition

118121 tr,

119122

123+ -- ** Scalar multiplication/division of matrices

124+ (.|),

125+ (./),

126+

120127 -- ** McCarthy's Conditional

121128 cond,

122129

123130 -- ** Matrix "abiding"

124− abideJS,

125− abideSJ,

131+ abideJF,

132+ abideFJ,

133+

134+ -- ** Zip Matrices

135+ zipWithM,

126136

127137 -- * Biproduct approach

128− -- ** Split

138+ -- ** Fork

129139 (===),

130140 -- *** Projections

131141 p1,

132142 p2,

133− -- ** Junc

143+ -- ** Join

134144 (|||),

135145 -- *** Injections

136146 i1,

@@ -144,11 +154,11 @@ module LAoP.Matrix.Type

144154 -- | Note that given the restrictions imposed it is not possible to

145155 -- implement the standard type classes present in standard Haskell.

146156 -- *** Matrix pairing projections

147− kp1,

148− kp2,

157+ fstM,

158+ sndM,

149159

150160 -- *** Matrix pairing

151− khatri,

161+ kr,

152162

153163 -- * Matrix composition and lifting

154164

@@ -156,10 +166,10 @@ module LAoP.Matrix.Type

156166

157167 -- | Note that given the restrictions imposed it is not possible to

158168 -- implement the standard type classes present in standard Haskell.

159− identity,

169+ iden,

160170 comp,

161− fromF,

162171 fromF',

172+ fromF,

163173

164174 -- * Relation

165175 toRel,

@@ -173,35 +183,44 @@ where

173183import Data.Void

174184import Data.Proxy

175185import Data.Kind

176−import GHC.TypeLits

186+import GHC.TypeLits

177187import Control.DeepSeq

178188import LAoP.Utils

179−import qualified Control.Category as C

180189import qualified LAoP.Matrix.Internal as I

190+import Prelude hiding (id, (.))

181191

182192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))

183193 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))

184194

185195-- | Constraint type synonyms to keep the type signatures less convoluted

186−type Countable a = KnownNat (I.Count a)

187−type CountableDimensions a b = (Countable a, Countable b)

188−type CountableN a = KnownNat (I.Count (I.Normalize a))

189−type CountableDimensionsN a b = (CountableN a, CountableN b)

190−type FromListsN e a b = I.FromLists e (I.Normalize a) (I.Normalize b)

191−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

192−type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))

193−type Trivial2 a = I.Normalize a ~ I.Normalize (I.Normalize a)

194−type Trivial3 a = I.FromNat (I.Count (I.Normalize (I.Normalize a))) ~ I.Normalize (I.Normalize a)

195−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

196−

197−-- | It isn't possible to implement the 'id' function so it's

198−-- implementation is 'undefined'. However 'comp' can be and this partial

199−-- class implementation exists just to make the code more readable.

200−--

201−-- Please use 'identity' instead.

202−instance (Num e) => C.Category (Matrix e) where

203− id = undefined

204− (.) = comp

196+type Countable a = KnownNat (I.Count a)

197+type CountableDims a b = (Countable a, Countable b)

198+type CountableN a = KnownNat (I.Count (I.Normalize a))

199+type CountableDimsN a b = (CountableN a, CountableN b)

200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)

201+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

202+type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))

203+type Trivial2 a = I.Normalize a ~ I.Normalize (I.Normalize a)

204+type Trivial3 a = I.FromNat (I.Count (I.Normalize (I.Normalize a))) ~ I.Normalize (I.Normalize a)

205+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

206+

207+-- | It is possible to implement a constrained version of the category type

208+-- class.

209+instance (Num e) => Category (Matrix e) where

210+ type Object (Matrix e) a = (FLN a a, CountableN a)

211+ id = iden

212+ (.) = comp

213+

214+-- | Bifunctor equivalent function

215+bimapM ::

216+ ( Liftable e a b,

217+ Liftable e c d,

218+ CountableDimsN a c,

219+ CountableDimsN b d,

220+ FLN d c,

221+ FLN b a

222+ ) => (a -> b) -> (c -> d) -> Matrix e a c -> Matrix e b d

223+bimapM f g m = fromF g . m . tr (fromF f)

205224

206225-- | Zero type alias

207226type Zero = Void

@@ -211,78 +230,74 @@ type One = ()

211230

212231-- Primitives

213232

214−-- | Empty matrix constructor

215−empty :: Matrix e Zero Zero

216−empty = M I.Empty

217−

218233-- | Unit matrix constructor

219234one :: e -> Matrix e One One

220235one = M . I.One

221236

222−-- | Matrix 'Junc' constructor

223−junc ::

237+-- | Matrix 'Join' constructor

238+join ::

224239 Matrix e a rows ->

225240 Matrix e b rows ->

226241 Matrix e (Either a b) rows

227−junc (M a) (M b) = M (I.Junc a b)

242+join (M a) (M b) = M (I.Join a b)

228243

229244infixl 3 |||

230−-- | Matrix 'Junc' constructor

245+-- | Matrix 'Join' constructor

231246(|||) ::

232247 Matrix e a rows ->

233248 Matrix e b rows ->

234249 Matrix e (Either a b) rows

235−(|||) = junc

250+(|||) = join

236251

237−-- | Matrix 'Split' constructor

238−split ::

252+-- | Matrix 'Fork' constructor

253+fork ::

239254 Matrix e cols a ->

240255 Matrix e cols b ->

241256 Matrix e cols (Either a b)

242−split (M a) (M b) = M (I.Split a b)

257+fork (M a) (M b) = M (I.Fork a b)

243258

244259infixl 2 ===

245−-- | Matrix 'Split' constructor

260+-- | Matrix 'Fork' constructor

246261(===) ::

247262 Matrix e cols a ->

248263 Matrix e cols b ->

249264 Matrix e cols (Either a b)

250−(===) = split

265+(===) = fork

251266

252267-- Functor hierarchy

253268

254269-- | Functor instance equivalent function

255−fmapM ::

270+fmapM ::

256271 ( Liftable e a b,

257− CountableDimensionsN a b,

258− FromListsN e b a

272+ CountableDimsN a b,

273+ FLN b a

259274 )

260275 =>

261276 (a -> b) -> Matrix e c a -> Matrix e c b

262−fmapM f m = fromF' f `comp` m

277+fmapM f m = fromF f . m

263278

264279-- | Applicative instance equivalent 'unit' function,

265280unitM :: (Num e) => Matrix e () ()

266281unitM = one 1

267282

268283-- | Applicative instance equivalent 'unit' function,

269−multM ::

270− ( CountableDimensionsN a b,

284+multM ::

285+ ( CountableDimsN a b,

271286 CountableN (a, b),

272287 Num e,

273− FromListsN e (a, b) a,

274− FromListsN e (a, b) b,

288+ FLN (a, b) a,

289+ FLN (a, b) b,

275290 TrivialP a b

276291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)

277−multM = khatri

292+multM = kr

278293

279294-- | Monad instance equivalent 'return' function,

280−returnM ::

281− forall e a .

295+returnM ::

296+ forall e a .

282297 ( Num e,

283298 Enum e,

284299 Enum a,

285− FromListsN e () a,

300+ FLN () a,

286301 Countable a

287302 ) => a -> Matrix e One a

288303returnM a = col l

@@ -299,51 +314,64 @@ bindM = flip comp

299314

300315-- | Build a matrix out of a list of list of elements. Throws a runtime

301316-- error if the dimensions do not match.

302−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows

317+fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows

303318fromLists = M . I.fromLists

304319

305320-- | Matrix builder function. Constructs a matrix provided with

306−-- a construction function.

307−matrixBuilder ::

308− (FromListsN e cols rows, CountableDimensionsN cols rows )

321+-- a construction function that operates with indices.

322+matrixBuilder' ::

323+ (FLN cols rows, CountableDimsN cols rows )

309324 => ((Int, Int) -> e) -> Matrix e cols rows

310−matrixBuilder = M . I.matrixBuilder

325+matrixBuilder' = M . I.matrixBuilder'

326+

327+-- | Matrix builder function. Constructs a matrix provided with

328+-- a construction function that operates with arbitrary types.

329+matrixBuilder ::

330+ ( FLN a b,

331+ Enum a,

332+ Enum b,

333+ Bounded a,

334+ Bounded b,

335+ Eq a,

336+ Countable b

337+ ) => ((a, b) -> e) -> Matrix e a b

338+matrixBuilder f = M (I.matrixBuilder f)

311339

312340-- | Constructs a column vector matrix

313−col :: (FromListsN e () rows) => [e] -> Matrix e One rows

341+col :: (FLN () rows) => [e] -> Matrix e One rows

314342col = M . I.col

315343

316344-- | Constructs a row vector matrix

317−row :: (FromListsN e cols ()) => [e] -> Matrix e cols One

345+row :: (FLN cols ()) => [e] -> Matrix e cols One

318346row = M . I.row

319347

320348-- | Lifts functions to matrices with arbitrary dimensions.

321349--

322350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

323351-- types @a@ or @b@ allows.

324−fromF ::

352+fromF' ::

325353 ( Liftable e a b,

326− CountableDimensionsN cols rows,

327− FromListsN e rows cols

354+ CountableDimsN cols rows,

355+ FLN rows cols

328356 ) =>

329357 (a -> b) -> Matrix e cols rows

330−fromF = M . I.fromF

358+fromF' = M . I.fromF'

331359

332360-- | Lifts functions to matrices with dimensions matching @a@ and @b@

333361-- cardinality's.

334−fromF' ::

362+fromF ::

335363 ( Liftable e a b,

336− CountableDimensionsN a b,

337− FromListsN e b a

364+ CountableDimsN a b,

365+ FLN b a

338366 ) =>

339367 (a -> b) -> Matrix e a b

340−fromF' = M . I.fromF'

368+fromF = M . I.fromF

341369

342370-- | Lifts relation functions to Boolean Matrix

343371toRel ::

344372 ( Liftable (Natural 0 1) a b,

345− CountableDimensionsN a b,

346− FromListsN (Natural 0 1) b a

373+ CountableDims a b,

374+ FLN b a

347375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b

348376toRel = M . I.toRel

349377

@@ -361,7 +389,7 @@ toList (M m) = I.toList m

361389

362390-- | The zero matrix. A matrix wholly filled with zeros.

363391zeros ::

364− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

392+ (Num e, FLN cols rows, CountableDimsN cols rows)

365393 => Matrix e cols rows

366394zeros = M I.zeros

367395

@@ -371,7 +399,7 @@ zeros = M I.zeros

371399--

372400-- Also known as T (Top) matrix.

373401ones ::

374− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

402+ (Num e, FLN cols rows, CountableDimsN cols rows)

375403 => Matrix e cols rows

376404ones = M I.ones

377405

@@ -380,7 +408,7 @@ ones = M I.ones

380408-- | The constant matrix constructor. A matrix wholly filled with a given

381409-- value.

382410constant ::

383− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

411+ (Num e, FLN cols rows, CountableDimsN cols rows)

384412 => e -> Matrix e cols rows

385413constant = M . I.constant

386414

@@ -389,30 +417,30 @@ constant = M . I.constant

389417-- | The T (Top) row vector matrix.

390418bang ::

391419 forall e cols.

392− (Num e, Enum e, FromListsN e cols (), CountableN cols) =>

420+ (Num e, Enum e, FLN cols (), CountableN cols) =>

393421 Matrix e cols One

394422bang = M I.bang

395423

396424-- | Point constant relation

397−point ::

425+point ::

398426 ( Bounded a,

399427 Enum a,

400428 Eq a,

401429 Num e,

402430 Ord e,

403431 CountableN a,

404− FromListsN e a One

432+ FLN a One

405433 ) => a -> Matrix e One a

406−point = fromF' . const

434+point = fromF . const

407435

408−-- Identity Matrix

436+-- iden Matrix

409437

410−-- | Identity matrix

411−identity ::

412− (Num e, FromListsN e a a, CountableN a) =>

438+-- | iden matrix

439+iden ::

440+ (Num e, FLN a a, CountableN a) =>

413441 Matrix e a a

414−identity = M I.identity

415−{-# NOINLINE identity #-}

442+iden = M I.iden

443+{-# NOINLINE iden #-}

416444

417445-- Matrix composition (MMM)

418446

@@ -423,17 +451,31 @@ identity = M I.identity

423451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

424452comp (M a) (M b) = M (I.comp a b)

425453{-# NOINLINE comp #-}

426−{-# RULES

427− "comp/identity1" forall m. comp m identity = m ;

428− "comp/identity2" forall m. comp identity m = m

454+{-# RULES

455+ "comp/iden1" forall m. comp m iden = m ;

456+ "comp/iden2" forall m. comp iden m = m

429457#-}

430458

459+-- Scalar multiplication of matrices

460+

461+infixl 7 .|

462+-- | Scalar multiplication of matrices.

463+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

464+(.|) e (M m) = M (e I..| m)

465+

466+-- Scalar division of matrices

467+

468+infixl 7 ./

469+-- | Scalar multiplication of matrices.

470+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

471+(./) (M m) e = M (m I../ e)

472+

431473-- | Biproduct first component projection

432474p1 ::

433475 ( Num e,

434− CountableDimensionsN n m,

435− FromListsN e n m,

436− FromListsN e m m

476+ CountableDimsN n m,

477+ FLN n m,

478+ FLN m m

437479 ) =>

438480 Matrix e (Either m n) m

439481p1 = M I.p1

@@ -441,9 +483,9 @@ p1 = M I.p1

441483-- | Biproduct second component projection

442484p2 ::

443485 ( Num e,

444− CountableDimensionsN n m,

445− FromListsN e m n,

446− FromListsN e n n

486+ CountableDimsN n m,

487+ FLN m n,

488+ FLN n n

447489 ) =>

448490 Matrix e (Either m n) n

449491p2 = M I.p2

@@ -453,9 +495,9 @@ p2 = M I.p2

453495-- | Biproduct first component injection

454496i1 ::

455497 ( Num e,

456− CountableDimensionsN n m,

457− FromListsN e n m,

458− FromListsN e m m

498+ CountableDimsN n m,

499+ FLN n m,

500+ FLN m m

459501 ) =>

460502 Matrix e m (Either m n)

461503i1 = tr p1

@@ -463,9 +505,9 @@ i1 = tr p1

463505-- | Biproduct second component injection

464506i2 ::

465507 ( Num e,

466− CountableDimensionsN n m,

467− FromListsN e m n,

468− FromListsN e n n

508+ CountableDimsN n m,

509+ FLN m n,

510+ FLN n n

469511 ) =>

470512 Matrix e n (Either m n)

471513i2 = tr p2

@@ -474,24 +516,34 @@ i2 = tr p2

474516

475517-- | Obtain the number of rows.

476518--

477−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

478−-- dimensions in constant time.

479−--

480−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

481−-- exchange for performance.

519+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

520+-- dimensions in constant time. For a version that doesn't require the

521+-- constraint see 'rows''.

482522rows :: (CountableN rows) => Matrix e cols rows -> Int

483523rows (M m) = I.rows m

484524

525+-- | Obtain the number of rows in an inefficient manner, but without any

526+-- constraints.

527+--

528+-- For a more efficient version see 'rows'.

529+rows' :: Matrix e cols rows -> Int

530+rows' (M m) = I.rows' m

531+

485532-- | Obtain the number of columns.

486−--

487−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

488−-- dimensions in constant time.

489533--

490−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

491−-- exchange for performance.

534+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

535+-- dimensions in constant time. For a version that doesn't require the

536+-- constraint see 'columns''.

492537columns :: (CountableN cols) => Matrix e cols rows -> Int

493538columns (M m) = I.columns m

494539

540+-- | Obtain the number of columns in an inefficient manner, but without any

541+-- constraints.

542+--

543+-- For a more efficient version see 'columns'.

544+columns' :: Matrix e cols rows -> Int

545+columns' (M m) = I.columns' m

546+

495547-- Coproduct Bifunctor

496548

497549infixl 5 -|-

@@ -499,11 +551,11 @@ infixl 5 -|-

499551-- | Matrix coproduct functor also known as matrix direct sum.

500552(-|-) ::

501553 ( Num e,

502− CountableDimensionsN j k,

503− FromListsN e k k,

504− FromListsN e j k,

505− FromListsN e k j,

506− FromListsN e j j

554+ CountableDimsN j k,

555+ FLN k k,

556+ FLN j k,

557+ FLN k j,

558+ FLN j j

507559 ) =>

508560 Matrix e n k ->

509561 Matrix e m j ->

@@ -513,98 +565,98 @@ infixl 5 -|-

513565-- Khatri Rao Product and projections

514566

515567-- | Khatri Rao product first component projection matrix.

516−kp1 ::

568+fstM ::

517569 forall e m k .

518570 ( Num e,

519− CountableDimensionsN m k,

571+ CountableDimsN m k,

520572 CountableN (m, k),

521− FromListsN e (m, k) m,

573+ FLN (m, k) m,

522574 TrivialP m k

523575 ) => Matrix e (m, k) m

524−kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))

576+fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))

525577

526578-- | Khatri Rao product second component projection matrix.

527−kp2 ::

579+sndM ::

528580 forall e m k.

529581 ( Num e,

530− CountableDimensionsN k m,

582+ CountableDimsN k m,

531583 CountableN (m, k),

532− FromListsN e (m, k) k,

584+ FLN (m, k) k,

533585 TrivialP m k

534586 ) => Matrix e (m, k) k

535−kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))

587+sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))

536588

537589-- | Khatri Rao Matrix product also known as matrix pairing.

538590--

539591-- NOTE: That this is not a true categorical product, see for instance:

540−--

592+--

541593-- @

542−-- | kp1 `comp` khatri a b == a

543−-- khatri a b ==> |

544−-- | kp2 `comp` khatri a b == b

594+-- | fstM . kr a b == a

595+-- kr a b ==> |

596+-- | sndM . kr a b == b

545597-- @

546598--

547599-- __Emphasis__ on the implication symbol.

548−khatri ::

600+kr ::

549601 forall e cols a b.

550602 ( Num e,

551− CountableDimensionsN a b,

603+ CountableDimsN a b,

552604 CountableN (a, b),

553− FromListsN e (a, b) a,

554− FromListsN e (a, b) b,

605+ FLN (a, b) a,

606+ FLN (a, b) b,

555607 TrivialP a b

556608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)

557−khatri a b =

558− let kp1' = kp1 @e @a @b

559− kp2' = kp2 @e @a @b

560− in comp (tr kp1') a * comp (tr kp2') b

609+kr a b =

610+ let fstM' = fstM @e @a @b

611+ sndM' = sndM @e @a @b

612+ in comp (tr fstM') a * comp (tr sndM') b

561613

562614-- Product Bifunctor (Kronecker)

563615

564616infixl 4 ><

565617

566−-- | Matrix product functor also known as kronecker product

618+-- | Matrix product functor also known as Kronecker product

567619(><) ::

568620 forall e m p n q.

569621 ( Num e,

570− CountableDimensionsN m n,

571− CountableDimensionsN p q,

572− CountableDimensionsN (m, n) (p, q),

573− FromListsN e (m, n) m,

574− FromListsN e (m, n) n,

575− FromListsN e (p, q) p,

576− FromListsN e (p, q) q,

622+ CountableDimsN m n,

623+ CountableDimsN p q,

624+ CountableDimsN (m, n) (p, q),

625+ FLN (m, n) m,

626+ FLN (m, n) n,

627+ FLN (p, q) p,

628+ FLN (p, q) q,

577629 TrivialP m n,

578630 TrivialP p q

579631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)

580632(><) a b =

581− let kp1' = kp1 @e @m @n

582− kp2' = kp2 @e @m @n

583− in khatri (comp a kp1') (comp b kp2')

633+ let fstM' = fstM @e @m @n

634+ sndM' = sndM @e @m @n

635+ in kr (comp a fstM') (comp b sndM')

584636

585−-- Matrix abide Junc Split

637+-- Matrix abide Join Fork

586638

587−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

588−--

639+-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.

640+--

589641-- Law:

590642--

591643-- @

592−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

644+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

593645-- @

594−abideJS :: Matrix e cols rows -> Matrix e cols rows

595−abideJS (M m) = M (I.abideJS m)

646+abideJF :: Matrix e cols rows -> Matrix e cols rows

647+abideJF (M m) = M (I.abideJF m)

596648

597−-- Matrix abide Split Junc

649+-- Matrix abide Fork Join

598650

599−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

600−--

651+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

652+--

601653-- Law:

602654--

603655-- @

604−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

656+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

605657-- @

606−abideSJ :: Matrix e cols rows -> Matrix e cols rows

607−abideSJ (M m) = M (I.abideSJ m)

658+abideFJ :: Matrix e cols rows -> Matrix e cols rows

659+abideFJ (M m) = M (I.abideFJ m)

608660

609661-- Matrix transposition

610662

@@ -616,9 +668,9 @@ tr (M m) = M (I.tr m)

616668

617669-- | Selective functors 'select' operator equivalent inspired by the

618670-- ArrowMonad solution presented in the paper.

619−selectM ::

671+selectM ::

620672 ( Num e,

621− FromListsN e b b,

673+ FLN b b,

622674 CountableN b

623675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

624676selectM (M m) (M y) = M (I.select m y)

@@ -631,9 +683,9 @@ cond ::

631683 Trivial2 a,

632684 Trivial3 a,

633685 CountableN a,

634− FromListsN e () a,

635− FromListsN e a (),

636− FromListsN e a a,

686+ FLN () a,

687+ FLN a (),

688+ FLN a a,

637689 Liftable e a Bool

638690 )

639691 =>

@@ -643,9 +695,13 @@ cond p (M a) (M b) = M (I.cond p a b)

643695-- Pretty print

644696

645697-- | Matrix pretty printer

646−pretty :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> String

698+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

647699pretty (M m) = I.pretty m

648700

649701-- | Matrix pretty printer

650−prettyPrint :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> IO ()

702+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

651703prettyPrint (M m) = I.prettyPrint m

704+

705+-- | Zip two matrices with a given binary function

706+zipWithM :: (e -> f -> g) -> Matrix e a b -> Matrix f a b -> Matrix g a b

707+zipWithM f (M a) (M b) = M (I.zipWithM f a b)

src/LAoP/Relation.hs+8−9

@@ -32,20 +32,19 @@ module LAoP.Relation

3232

3333 -- * Constraint type synonyms

3434 Countable,

35− CountableDimensions,

35+ CountableDims,

3636 CountableN,

37− CountableDimensionsN,

38− FromListsN,

37+ CountableDimsN,

38+ FLN,

3939 Liftable,

4040 Trivial,

4141 TrivialP,

4242

4343 -- * Primitives

44− empty,

4544 one,

46− junc,

45+ join,

4746 (|||),

48− split,

47+ fork,

4948 (===),

5049

5150 -- * Auxiliary type families

@@ -108,8 +107,8 @@ module LAoP.Relation

108107 splitR,

109108

110109 -- ** Projections

111− p1,

112− p2,

110+ fstR,

111+ sndR,

113112 -- ** Bifunctor

114113 (><),

115114 -- * Relational coproduct

@@ -148,7 +147,7 @@ module LAoP.Relation

148147 cond,

149148

150149 -- * Relational composition and lifting

151− identity,

150+ iden,

152151 comp,

153152 fromF,

154153 fromF',

src/LAoP/Relation/Internal.hs+245−234

@@ -1,3 +1,4 @@

1+{-# LANGUAGE TypeFamilies #-}

12{-# LANGUAGE ConstraintKinds #-}

23{-# LANGUAGE GADTs #-}

34{-# LANGUAGE UndecidableInstances #-}

@@ -42,20 +43,19 @@ module LAoP.Relation.Internal

4243

4344 -- * Constraint type synonyms

4445 Countable,

45− CountableDimensions,

46+ CountableDims,

4647 CountableN,

47− CountableDimensionsN,

48− FromListsN,

48+ CountableDimsN,

49+ FLN,

4950 Liftable,

5051 Trivial,

5152 TrivialP,

5253

5354 -- * Primitives

54− empty,

5555 one,

56− junc,

56+ join,

5757 (|||),

58− split,

58+ fork,

5959 (===),

6060

6161 -- * Auxiliary type families

@@ -66,14 +66,15 @@ module LAoP.Relation.Internal

6666 -- * Matrix construction and conversion

6767 I.FromLists,

6868 fromLists,

69− fromF,

7069 fromF',

70+ fromF,

7171 toRel,

7272 toLists,

7373 toList,

7474 toBool,

7575 pt,

7676 belongs,

77+ relationBuilder',

7778 relationBuilder,

7879 zeros,

7980 ones,

@@ -118,8 +119,8 @@ module LAoP.Relation.Internal

118119 splitR,

119120

120121 -- ** Projections

121− p1,

122− p2,

122+ fstR,

123+ sndR,

123124 -- ** Bifunctor

124125 (><),

125126 -- * Relational coproduct

@@ -158,10 +159,10 @@ module LAoP.Relation.Internal

158159 cond,

159160

160161 -- * Relational composition and lifting

161− identity,

162+ iden,

162163 comp,

163− fromF,

164164 fromF',

165+ fromF,

165166

166167 -- ** Relational application

167168 pointAp,

@@ -177,9 +178,9 @@ import Data.Void

177178import qualified LAoP.Matrix.Internal as I

178179import LAoP.Utils.Internal

179180import Control.DeepSeq

180−import qualified Control.Category as C

181181import Data.Bool

182182import GHC.TypeLits

183+import Prelude hiding (id, (.))

183184

184185-- | Boolean type synonym for working with boolean matrices

185186type Boolean = Natural 0 1

@@ -192,32 +193,30 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

192193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)

193194

194195-- | Constraint type synonyms to keep the type signatures less convoluted

195−type Countable a = KnownNat (I.Count a)

196−type CountableDimensions a b = (Countable a, Countable b)

197−type CountableN a = KnownNat (I.Count (I.Normalize a))

198−type CountableDimensionsN a b = (CountableN a, CountableN b)

199−type FromListsN a b = I.FromLists Boolean (I.Normalize a) (I.Normalize b)

200−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)

201−type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)

202−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

203−

204−-- | It isn't possible to implement the 'id' function so it's

205−-- implementation is 'undefined'. However 'comp' can be and this partial

206−-- class implementation exists just to make the code more readable.

207−--

208−-- Please use 'identity' instead.

209−instance C.Category Relation where

210− id = undefined

211− (.) = comp

196+type Countable a = KnownNat (I.Count a)

197+type CountableDims a b = (Countable a, Countable b)

198+type CountableN a = KnownNat (I.Count (I.Normalize a))

199+type CountableDimsN a b = (CountableN a, CountableN b)

200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)

201+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)

202+type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)

203+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

204+

205+-- | It is possible to implement a constrained version of the category type

206+-- class.

207+instance Category Relation where

208+ type Object Relation a = (FLN a a, CountableN a)

209+ id = iden

210+ (.) = comp

212211

213212instance Num (Relation a b) where

214− -- | Matrix addition becomes Boolean matrix disjunction

213+ -- | Matrix addition becomes Boolean matrix disjointion

215214 (R a) + (R b) = R (I.orM a b)

216215

217216 -- | Matrix subtraction becomes Relational subtraction

218217 (R a) - (R b) = R (I.subM a b)

219218

220− -- | Matrix multiplication becomes Boolean matrix conjunction

219+ -- | Matrix multiplication becomes Boolean matrix conjointion

221220 (R a) * (R b) = R (I.andM a b)

222221

223222 -- | Matrix negation becomes Boolean matrix negation

@@ -229,83 +228,92 @@ type One = ()

229228

230229-- Primitives

231230

232−-- | Empty matrix constructor

233−empty :: Relation Zero Zero

234−empty = R I.Empty

235−

236231-- | Unit matrix constructor

237232one :: Boolean -> Relation One One

238233one = R . I.One

239234

240−-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.

235+-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.

241236--

242237-- See 'eitherR'.

243−junc :: Relation a c -> Relation b c -> Relation (Either a b) c

244−junc (R a) (R b) = R (I.Junc a b)

238+join :: Relation a c -> Relation b c -> Relation (Either a b) c

239+join (R a) (R b) = R (I.Join a b)

245240

246241infixl 3 |||

247−-- | Boolean Matrix 'Junc' constructor

242+-- | Boolean Matrix 'Join' constructor

248243--

249244-- See 'eitherR'.

250245(|||) ::

251246 Relation a c ->

252247 Relation b c ->

253248 Relation (Either a b) c

254−(|||) = junc

249+(|||) = join

255250

256−-- | Boolean Matrix 'Split' constructor, also known as relational product.

257−split :: Relation c a -> Relation c b -> Relation c (Either a b)

258−split (R a) (R b) = R (I.Split a b)

251+-- | Boolean Matrix 'Fork' constructor, also known as relational product.

252+fork :: Relation c a -> Relation c b -> Relation c (Either a b)

253+fork (R a) (R b) = R (I.Fork a b)

259254

260255infixl 2 ===

261−-- | Boolean Matrix 'Split' constructor

256+-- | Boolean Matrix 'Fork' constructor

262257(===) ::

263258 Relation c a ->

264259 Relation c b ->

265260 Relation c (Either a b)

266−(===) = split

261+(===) = fork

267262

268263-- Construction

269264

270265-- | Build a matrix out of a list of list of elements. Throws a runtime

271266-- error if the dimensions do not match.

272−fromLists :: (FromListsN a b) => [[Boolean]] -> Relation a b

267+fromLists :: (FLN a b) => [[Boolean]] -> Relation a b

273268fromLists = R . I.fromLists

274269

275−-- | Matrix builder function. Constructs a matrix provided with

276−-- a construction function.

277−relationBuilder ::

278− (FromListsN a b, CountableDimensionsN a b) =>

270+-- | Relation builder function. Constructs a relation provided with

271+-- a construction function that operates with indices.

272+relationBuilder' ::

273+ (FLN a b, CountableDimsN a b) =>

279274 ((Int, Int) -> Boolean) -> Relation a b

275+relationBuilder' = R . I.matrixBuilder'

276+

277+-- | Relation builder function. Constructs a relation provided with

278+-- a construction function that operates with arbitrary types.

279+relationBuilder ::

280+ ( FLN a b,

281+ Enum a,

282+ Enum b,

283+ Bounded a,

284+ Bounded b,

285+ Eq a,

286+ CountableDims a b

287+ ) => ((a, b) -> Boolean) -> Relation a b

280288relationBuilder = R . I.matrixBuilder

281289

282290-- | Lifts functions to matrices with arbitrary dimensions.

283291--

284292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

285293-- types @a@ or @b@ allows.

286−fromF ::

294+fromF' ::

287295 ( Liftable a b,

288− CountableDimensionsN c d,

289− FromListsN d c

296+ CountableDimsN c d,

297+ FLN d c

290298 )

291299 => (a -> b) -> Relation c d

292−fromF f = R (I.fromFRel f)

300+fromF' f = R (I.fromFRel' f)

293301

294302-- | Lifts functions to matrices with dimensions matching @a@ and @b@

295303-- cardinality's.

296−fromF' ::

304+fromF ::

297305 ( Liftable a b,

298− CountableDimensionsN a b,

299− FromListsN b a

306+ CountableDimsN a b,

307+ FLN b a

300308 )

301309 => (a -> b) -> Relation a b

302−fromF' f = R (I.fromFRel' f)

310+fromF f = R (I.fromFRel f)

303311

304312-- | Lifts relation functions to 'Relation'

305313toRel ::

306314 ( Liftable a b,

307− CountableDimensionsN a b,

308− FromListsN b a

315+ CountableDims a b,

316+ FLN b a

309317 )

310318 => (a -> b -> Bool) -> Relation a b

311319toRel = R . I.toRel

@@ -314,9 +322,9 @@ toRel = R . I.toRel

314322fromRel ::

315323 ( Liftable a b,

316324 Eq a,

317− CountableDimensionsN a b,

318− FromListsN a One,

319− FromListsN b One

325+ CountableDimsN a b,

326+ FLN a One,

327+ FLN b One

320328 )

321329 => Relation a b -> (a -> b -> Bool)

322330fromRel r a b = pointApBool a b r

@@ -340,12 +348,12 @@ toBool r = case toList r of

340348-- | Power transpose.

341349--

342350-- Maps a relation to a set valued function.

343−pt ::

351+pt ::

344352 ( Liftable a b,

345353 Eq a,

346− CountableDimensionsN a b,

347− FromListsN a One,

348− FromListsN b One

354+ CountableDimsN a b,

355+ FLN a One,

356+ FLN b One

349357 )

350358 => Relation a b -> (a -> List b)

351359pt r a =

@@ -357,8 +365,8 @@ belongs ::

357365 ( Bounded a,

358366 Enum a,

359367 Eq a,

360− CountableDimensionsN (List a) a,

361− FromListsN a (List a)

368+ CountableDims (List a) a,

369+ FLN a (List a)

362370 )

363371 => Relation (List a) a

364372belongs = toRel elemR

@@ -370,16 +378,16 @@ belongs = toRel elemR

370378-- | The zero relation. A relation where no element of type @a@ relates

371379-- with elements of type @b@.

372380--

373−-- Also known as ⊥ (Bottom) Relation or empty Relation.

381+-- Also known as ⊥ (Bottom) Relation.

374382--

375383-- @

376−-- r ``comp`` ⊥ == ⊥ ``comp`` r == ⊥

384+-- r `.` ⊥ == ⊥ `.` r == ⊥

377385-- ⊥ ``sse`` R && R ``sse`` T == True

378386-- @

379387zeros ::

380− (FromListsN a b, CountableDimensionsN a b) =>

388+ (FLN a b, CountableDimsN a b) =>

381389 Relation a b

382−zeros = relationBuilder (const (nat 0))

390+zeros = relationBuilder' (const (nat 0))

383391

384392-- Ones Matrix

385393

@@ -392,43 +400,43 @@ zeros = relationBuilder (const (nat 0))

392400-- ⊥ ``sse`` R && R ``sse`` T == True

393401-- @

394402ones ::

395− (FromListsN a b, CountableDimensionsN a b) =>

403+ (FLN a b, CountableDimsN a b) =>

396404 Relation a b

397−ones = relationBuilder (const (nat 1))

405+ones = relationBuilder' (const (nat 1))

398406

399407-- Bang Matrix

400408

401409-- | The T (Top) row vector relation.

402410bang ::

403− (FromListsN a One, CountableN a) =>

411+ (FLN a One, CountableN a) =>

404412 Relation a One

405413bang = ones

406414

407415-- | Point constant relation

408−point ::

416+point ::

409417 ( Bounded a,

410418 Enum a,

411419 Eq a,

412420 CountableN a,

413− FromListsN a One

421+ FLN a One

414422 ) => a -> Relation One a

415−point = fromF' . const

423+point = fromF . const

416424

417−-- Identity Matrix

425+-- iden Matrix

418426

419−-- | Identity matrix

427+-- | iden matrix

420428--

421429-- @

422−-- 'identity' ``comp`` r == r == r ``comp`` 'identity'

430+-- 'iden' `.` r == r == r `.` 'iden'

423431-- @

424−identity ::

425− (FromListsN a a, CountableN a) => Relation a a

426−identity = relationBuilder (bool (nat 0) (nat 1) . uncurry (==))

432+iden ::

433+ (FLN a a, CountableN a) => Relation a a

434+iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))

427435

428436-- | Relational composition

429437--

430438-- @

431−-- r ``comp`` (s ``comp`` p) = (r ``comp`` s) ``comp`` p

439+-- r `.` (s `.` p) = (r `.` s) `.` p

432440-- @

433441comp :: Relation b c -> Relation a b -> Relation a c

434442comp (R a) (R b) = R (I.compRel a b)

@@ -476,7 +484,7 @@ shrunkBy r s = r `intersection` divR s (conv r)

476484-- r ``overriddenBy`` r == r

477485-- @

478486overriddenBy ::

479− ( FromListsN b b,

487+ ( FLN b b,

480488 CountableN b

481489 ) => Relation a b -> Relation a b -> Relation a b

482490overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

@@ -488,11 +496,11 @@ overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

488496pointAp ::

489497 ( Liftable a b,

490498 Eq a,

491− CountableDimensionsN a b,

492− FromListsN a One,

493− FromListsN b One

499+ CountableDimsN a b,

500+ FLN a One,

501+ FLN b One

494502 ) => a -> b -> Relation a b -> Relation One One

495−pointAp a b r = conv (point b) `comp` r `comp` point a

503+pointAp a b r = conv (point b) . r . point a

496504

497505-- | Relational application

498506--

@@ -500,11 +508,11 @@ pointAp a b r = conv (point b) `comp` r `comp` point a

500508pointApBool ::

501509 ( Liftable a b,

502510 Eq a,

503− CountableDimensionsN a b,

504− FromListsN a One,

505− FromListsN b One

511+ CountableDimsN a b,

512+ FLN a One,

513+ FLN b One

506514 ) => a -> b -> Relation a b -> Bool

507−pointApBool a b r = toBool $ conv (point b) `comp` r `comp` point a

515+pointApBool a b r = toBool $ conv (point b) . r . point a

508516

509517-- | Relational converse

510518--

@@ -530,7 +538,7 @@ iff r s = r == s

530538

531539-- | Relational intersection

532540--

533−-- Lifts pointwise conjunction.

541+-- Lifts pointwise conjointion.

534542--

535543-- @

536544-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)

@@ -541,13 +549,13 @@ intersection a b = a * b

541549

542550-- | Relational union

543551--

544−-- Lifts pointwise disjunction.

552+-- Lifts pointwise disjointion.

545553--

546554-- @

547555-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)

548556-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x

549−-- r ``comp`` (s ``union`` t) == (r ``comp`` s) ``union`` (r ``comp`` t)

550−-- (s ``union`` t) ``comp`` r == (s ``comp`` r) ``union`` (t ``comp`` r)

557+-- r `.` (s ``union`` t) == (r `.` s) ``union`` (r `.` t)

558+-- (s ``union`` t) `.` r == (s `.` r) ``union`` (t `.` r)

551559-- @

552560union :: Relation a b -> Relation a b -> Relation a b

553561union a b = a + b

@@ -555,46 +563,46 @@ union a b = a + b

555563-- | Relation Kernel

556564--

557565-- @

558−-- 'ker' r == 'conv' r ``comp`` r

566+-- 'ker' r == 'conv' r `.` r

559567-- 'ker' r == 'img' ('conv' r)

560568-- @

561569ker :: Relation a b -> Relation a a

562−ker r = conv r `comp` r

570+ker r = conv r . r

563571

564572-- | Relation Image

565573--

566574-- @

567−-- 'img' r == r ``comp`` conv r

575+-- 'img' r == r `.` conv r

568576-- 'img' r == 'ker' ('conv' r)

569577-- @

570578img :: Relation a b -> Relation b b

571−img r = r `comp` conv r

579+img r = r . conv r

572580

573581-- | Function division. Special case of 'divS'.

574582--

575583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and

576584-- 'entire'.

577585--

578−-- @'divisionF' f g == 'conv' g ``comp`` f@

586+-- @'divisionF' f g == 'conv' g `.` f@

579587divisionF :: Relation a c -> Relation b c -> Relation a b

580−divisionF f g = conv g `comp` f

588+divisionF f g = conv g . f

581589

582590-- Taxonomy of binary relations

583591

584592-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@

585−simple :: (CountableN b, FromListsN b b) => Relation a b -> Bool

593+simple :: (CountableN b, FLN b b) => Relation a b -> Bool

586594simple = coreflexive . img

587595

588596-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@

589−injective :: (CountableN a, FromListsN a a) => Relation a b -> Bool

597+injective :: (CountableN a, FLN a a) => Relation a b -> Bool

590598injective = coreflexive . ker

591599

592600-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@

593−entire :: (CountableN a, FromListsN a a) => Relation a b -> Bool

601+entire :: (CountableN a, FLN a a) => Relation a b -> Bool

594602entire = reflexive . ker

595603

596604-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@

597−surjective :: (CountableN b, FromListsN b b) => Relation a b -> Bool

605+surjective :: (CountableN b, FLN b b) => Relation a b -> Bool

598606surjective = reflexive . img

599607

600608-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@

@@ -603,13 +611,13 @@ surjective = reflexive . img

603611-- relations:

604612--

605613-- @

606−-- f ``comp`` r ``sse`` s == r ``sse`` f ``comp`` s

607−-- r ``comp`` f ``sse`` s == r ``sse`` s ``comp`` f

614+-- f `.` r ``sse`` s == r ``sse`` f `.` s

615+-- r `.` f ``sse`` s == r ``sse`` s `.` f

608616-- @

609617function ::

610− ( CountableDimensionsN a b,

611− FromListsN a a,

612− FromListsN b b

618+ ( CountableDimsN a b,

619+ FLN a a,

620+ FLN b b

613621 )

614622 => Relation a b -> Bool

615623function r = simple r && entire r

@@ -617,7 +625,7 @@ function r = simple r && entire r

617625-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@

618626representation ::

619627 ( CountableN a,

620− FromListsN a a

628+ FLN a a

621629 )

622630 => Relation a b -> Bool

623631representation r = injective r && entire r

@@ -625,91 +633,91 @@ representation r = injective r && entire r

625633-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@

626634abstraction ::

627635 ( CountableN b,

628− FromListsN b b

636+ FLN b b

629637 )

630638 => Relation a b -> Bool

631639abstraction r = surjective r && simple r

632640

633641-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@

634642surjection ::

635− ( CountableDimensionsN a b,

636− FromListsN a a,

637− FromListsN b b

643+ ( CountableDimsN a b,

644+ FLN a a,

645+ FLN b b

638646 )

639647 => Relation a b -> Bool

640648surjection r = function r && abstraction r

641649

642650-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@

643651injection ::

644− ( CountableDimensionsN a b,

645− FromListsN a a,

646− FromListsN b b

652+ ( CountableDimsN a b,

653+ FLN a a,

654+ FLN b b

647655 )

648656 => Relation a b -> Bool

649657injection r = function r && representation r

650658

651659-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@

652660bijection ::

653− ( CountableDimensionsN a b,

654− FromListsN b b,

655− FromListsN a a

661+ ( CountableDimsN a b,

662+ FLN b b,

663+ FLN a a

656664 ) => Relation a b -> Bool

657665bijection r = injection r && surjection r

658666

659667-- Properties of relations

660668

661−-- | A 'Relation' @r@ is 'reflexive' 'iff' @'identity' ``sse`` r@

662−reflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

663−reflexive r = identity <= r

669+-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@

670+reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

671+reflexive r = id <= r

664672

665−-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'identity'@

666−coreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

667−coreflexive r = r <= identity

673+-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@

674+coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

675+coreflexive r = r <= id

668676

669−-- | A 'Relation' @r@ is 'transitive' 'iff' @(r ``comp`` r) ``sse`` r@

677+-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@

670678transitive :: Relation a a -> Bool

671−transitive r = (r `comp` r) `sse` r

679+transitive r = (r . r) `sse` r

672680

673681-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@

674682symmetric :: Relation a a -> Bool

675683symmetric r = r == conv r

676684

677−-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'identity'@

678−antiSymmetric :: (CountableN a, FromListsN a a) => Relation a a -> Bool

679−antiSymmetric r = (r `intersection` conv r) `sse` identity

685+-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@

686+antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool

687+antiSymmetric r = (r `intersection` conv r) `sse` id

680688

681−-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'identity') == 'zeros'@

682−irreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

683−irreflexive r = (r `intersection` identity) == zeros

689+-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@

690+irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

691+irreflexive r = (r `intersection` id) == zeros

684692

685693-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@

686−connected :: (CountableN a, FromListsN a a) => Relation a a -> Bool

694+connected :: (CountableN a, FLN a a) => Relation a a -> Bool

687695connected r = (r `union` conv r) == ones

688696

689697-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@

690−preorder :: (CountableN a, FromListsN a a) => Relation a a -> Bool

698+preorder :: (CountableN a, FLN a a) => Relation a a -> Bool

691699preorder r = reflexive r && transitive r

692700

693701-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@

694−partialOrder :: (CountableN a, FromListsN a a) => Relation a a -> Bool

702+partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

695703partialOrder r = antiSymmetric r && preorder r

696704

697705-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@

698−linearOrder :: (CountableN a, FromListsN a a) => Relation a a -> Bool

706+linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

699707linearOrder r = connected r && partialOrder r

700708

701709-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@

702−equivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool

710+equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

703711equivalence r = symmetric r && preorder r

704712

705713-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@

706−partialEquivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool

714+partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

707715partialEquivalence r = partialOrder r && equivalence r

708716

709717-- | A 'Relation' @r@ is 'difunctional' or regular wherever

710−-- @r ``comp`` 'conv' r ``comp`` r == r@

718+-- @r `.` 'conv' r `.` r == r@

711719difunctional :: Relation a b -> Bool

712−difunctional r = r `comp` conv r `comp` r == r

720+difunctional r = r . conv r . r == r

713721

714722-- Relational pairing

715723

@@ -718,61 +726,61 @@ difunctional r = r `comp` conv r `comp` r == r

718726-- NOTE: That this is not a true categorical product, see for instance:

719727--

720728-- @

721−-- | 'p1' ``comp`` 'splitR' a b ``sse`` a

729+-- | 'fstR' `.` 'splitR' a b ``sse`` a

722730-- 'splitR' a b <=> |

723−-- | 'p2' ``comp`` 'splitR' a b ``sse`` b

731+-- | 'sndR' `.` 'splitR' a b ``sse`` b

724732-- @

725733--

726734-- __Emphasis__ on the 'sse'.

727735--

728736-- @

729−-- 'splitR' r s ``comp`` f == 'splitR' (r ``comp`` f) (s ``comp`` f)

730−-- (R '><' S) ``comp`` 'splitR' p q == 'splitR' (r ``comp`` p) (s ``comp`` q)

731−-- 'conv' ('splitR' r s) ``comp`` 'splitR' x y == ('conv' r ``comp`` x) ``intersection`` ('conv' s ``comp`` y)

737+-- 'splitR' r s `.` f == 'splitR' (r `.` f) (s `.` f)

738+-- (R '><' S) `.` 'splitR' p q == 'splitR' (r `.` p) (s `.` q)

739+-- 'conv' ('splitR' r s) `.` 'splitR' x y == ('conv' r `.` x) ``intersection`` ('conv' s `.` y)

732740-- @

733741--

734742-- @

735743-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

736744-- @

737745splitR ::

738− ( CountableDimensionsN a b,

746+ ( CountableDimsN a b,

739747 CountableN (a, b),

740− FromListsN (a, b) a,

741− FromListsN (a, b) b,

748+ FLN (a, b) a,

749+ FLN (a, b) b,

742750 TrivialP a b

743751 )

744752 => Relation c a -> Relation c b -> Relation c (a, b)

745−splitR (R f) (R g) = R (I.khatri f g)

753+splitR (R f) (R g) = R (I.kr f g)

746754

747755-- | Relational pairing first component projection

748756--

749757-- @

750−-- 'p1' ``comp`` 'splitR' r s ``sse`` r

758+-- 'fstR' `.` 'splitR' r s ``sse`` r

751759-- @

752−p1 ::

760+fstR ::

753761 forall a b .

754− ( CountableDimensionsN a b,

762+ ( CountableDimsN a b,

755763 CountableN (a, b),

756− FromListsN (a, b) a,

764+ FLN (a, b) a,

757765 TrivialP a b

758766 )

759767 => Relation (a, b) a

760−p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))

768+fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))

761769

762770-- | Relational pairing second component projection

763771--

764772-- @

765−-- 'p2' ``comp`` 'splitR' r s ``sse`` s

773+-- 'sndR' `.` 'splitR' r s ``sse`` s

766774-- @

767−p2 ::

775+sndR ::

768776 forall a b .

769− ( CountableDimensionsN a b,

777+ ( CountableDimsN a b,

770778 CountableN (a, b),

771− FromListsN (a, b) b,

779+ FLN (a, b) b,

772780 TrivialP a b

773781 )

774782 => Relation (a, b) b

775−p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))

783+sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))

776784

777785-- Relational pairing functor

778786

@@ -780,17 +788,17 @@ infixl 4 ><

780788-- | Relational pairing functor

781789--

782790-- @

783−-- r '><' s == 'splitR' (r ``comp`` p1) (s ``comp`` p2)

784−-- (r '><' s) ``comp`` (p '><' q) == (r ``comp`` p) '><' (s ``comp`` q)

791+-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)

792+-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)

785793-- @

786794(><) ::

787− ( CountableDimensionsN a b,

788− CountableDimensionsN c d,

789− CountableDimensionsN (a, c) (b, d),

790− FromListsN (a, c) a,

791− FromListsN (a, c) c,

792− FromListsN (b, d) b,

793− FromListsN (b, d) d,

795+ ( CountableDimsN a b,

796+ CountableDimsN c d,

797+ CountableDimsN (a, c) (b, d),

798+ FLN (a, c) a,

799+ FLN (a, c) c,

800+ FLN (b, d) b,

801+ FLN (b, d) d,

794802 TrivialP a c,

795803 TrivialP b d

796804 )

@@ -802,31 +810,31 @@ infixl 4 ><

802810-- | Relational coproduct.

803811--

804812-- @

805−-- | 'eitherR' a b ``comp`` 'i1' == a

813+-- | 'eitherR' a b `.` 'i1' == a

806814-- 'eitherR' a b <=> |

807−-- | 'eitherR' a b ``comp`` 'i2' == b

815+-- | 'eitherR' a b `.` 'i2' == b

808816-- @

809817--

810818-- @

811−-- 'eitherR' r s ``comp`` 'conv' ('eitherR' t u) == (r ``comp`` 'conv' t) ``union`` (s ``comp`` 'conv' u)

819+-- 'eitherR' r s `.` 'conv' ('eitherR' t u) == (r `.` 'conv' t) ``union`` (s `.` 'conv' u)

812820-- @

813821--

814822-- @

815823-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

816824-- @

817825eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c

818−eitherR = junc

826+eitherR = join

819827

820828-- | Relational coproduct first component injection

821829--

822830-- @

823−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'

824−-- 'i1' ``comp`` 'i2' = 'zeros'

831+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

832+-- 'i1' `.` 'i2' = 'zeros'

825833-- @

826834i1 ::

827− ( CountableDimensionsN a b,

828− FromListsN b a,

829− FromListsN a a

835+ ( CountableDimsN a b,

836+ FLN b a,

837+ FLN a a

830838 )

831839 => Relation a (Either a b)

832840i1 = R I.i1

@@ -834,13 +842,13 @@ i1 = R I.i1

834842-- | Relational coproduct second component injection

835843--

836844-- @

837−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'

838−-- 'i1' ``comp`` 'i2' = 'zeros'

845+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

846+-- 'i1' `.` 'i2' = 'zeros'

839847-- @

840848i2 ::

841− ( CountableDimensionsN a b,

842− FromListsN a b,

843− FromListsN b b

849+ ( CountableDimsN a b,

850+ FLN a b,

851+ FLN b b

844852 )

845853 => Relation b (Either a b)

846854i2 = R I.i2

@@ -850,14 +858,14 @@ infixl 5 -|-

850858-- | Relational coproduct functor.

851859--

852860-- @

853−-- r '-|-' s == 'eitherR' ('i1' ``comp`` r) ('i2' ``comp`` s)

861+-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)

854862-- @

855863(-|-) ::

856− ( CountableDimensionsN b d,

857− FromListsN b b,

858− FromListsN d b,

859− FromListsN b d,

860− FromListsN d d

864+ ( CountableDimsN b d,

865+ FLN b b,

866+ FLN d b,

867+ FLN b d,

868+ FLN d d

861869 )

862870 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)

863871(-|-) (R a) (R b) = R ((I.-|-) a b)

@@ -870,20 +878,20 @@ infixl 5 -|-

870878-- 'trans'/'untrans';

871879-- more-over, where each particular attribute is placed (input/output) is irrelevant.

872880trans ::

873− ( CountableDimensionsN a b,

881+ ( CountableDimsN a b,

874882 CountableN c,

875− CountableDimensionsN (a, b) (c, b),

876− FromListsN (c, b) c,

877− FromListsN (c, b) b,

878− FromListsN (a, b) a,

879− FromListsN (a, b) b,

883+ CountableDimsN (a, b) (c, b),

884+ FLN (c, b) c,

885+ FLN (c, b) b,

886+ FLN (a, b) a,

887+ FLN (a, b) b,

880888 Trivial (a, b),

881889 Trivial (c, b),

882890 TrivialP a b,

883891 TrivialP c b

884892 )

885893 => Relation (a, b) c -> Relation a (c, b)

886−trans r = splitR r p2 `comp` conv p1

894+trans r = splitR r sndR . conv fstR

887895

888896-- | Relational 'untrans'

889897--

@@ -891,107 +899,110 @@ trans r = splitR r p2 `comp` conv p1

891899-- 'trans'/'untrans';

892900-- more-over, where each particular attribute is placed (input/output) is irrelevant.

893901untrans ::

894− ( CountableDimensionsN a b,

902+ ( CountableDimsN a b,

895903 CountableN c,

896− CountableDimensionsN (a, b) (c, b),

897− FromListsN (c, b) c,

898− FromListsN (c, b) b,

899− FromListsN (a, b) b,

900− FromListsN (a, b) a,

904+ CountableDimsN (a, b) (c, b),

905+ FLN (c, b) c,

906+ FLN (c, b) b,

907+ FLN (a, b) b,

908+ FLN (a, b) a,

901909 Trivial (a, b),

902910 Trivial (c, b),

903911 TrivialP a b,

904912 TrivialP c b

905913 )

906914 => Relation a (c, b) -> Relation (a, b) c

907−untrans s = p1 `comp` conv (splitR (conv s) p2)

915+untrans s = fstR . conv (splitR (conv s) sndR)

908916

909917-- | Transforms predicate @p@ into a correflexive relation.

910918--

911919-- @

912−-- 'predR' ('const' True) == 'identity'

920+-- 'predR' ('const' True) == 'id'

913921-- 'predR' ('const' False) == 'zeros'

914922-- @

915923--

916924-- @

917−-- 'predR' q ``comp`` 'predR' p == 'predR' q ``union`` 'predR' p

925+-- 'predR' q `.` 'predR' p == 'predR' q ``union`` 'predR' p

918926-- @

919927predR ::

920928 ( Bounded a,

921929 Enum a,

930+ Countable a,

922931 CountableN a,

923− FromListsN a a,

924− FromListsN Bool a

932+ FLN a a,

933+ FLN Bool a

925934 )

926935 => Relation a Bool -> Relation a a

927−predR p = identity `intersection` divisionF (fromF' (const True)) p

936+predR p = id `intersection` divisionF (fromF (const True)) p

928937

929938-- | Equalizes functions @f@ and @g@.

930939-- That is, @'equalizer' f g@ is the largest coreflexive

931940-- that restricts @g@ so that @f@ and @g@ yield the same outputs.

932941--

933942-- @

934−-- 'equalizer' r r == 'identity'

943+-- 'equalizer' r r == 'id'

935944-- 'equalizer' ('point' True) ('point' False) == 'zeros'

936945-- @

937946equalizer ::

938947 ( CountableN a,

939− FromListsN a a

948+ FLN a a

940949 )

941950 => Relation a b -> Relation a b -> Relation a a

942−equalizer f g = identity `intersection` divisionF f g

951+equalizer f g = id `intersection` divisionF f g

943952

944953-- | Relational conditional guard.

945954--

946955-- @

947−-- 'guard' p = 'i2' ``overriddenBy`` 'i1' ``comp`` 'predR' p

956+-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p

948957-- @

949−guard ::

958+guard ::

950959 ( Bounded b,

951960 Enum b,

952961 CountableN b,

953− FromListsN b b,

954− FromListsN Bool b

962+ Countable b,

963+ FLN b b,

964+ FLN Bool b

955965 ) => Relation b Bool -> Relation b (Either b b)

956966guard p = conv (eitherR (predR p) (predR (negate p)))

957967

958968-- | Relational McCarthy's conditional.

959−cond ::

969+cond ::

960970 ( Bounded b,

961971 Enum b,

972+ Countable b,

962973 CountableN b,

963− FromListsN b b,

964− FromListsN Bool b

965− )

974+ FLN b b,

975+ FLN Bool b

976+ )

966977 => Relation b Bool -> Relation b c -> Relation b c -> Relation b c

967−cond p r s = eitherR r s `comp` guard p

978+cond p r s = eitherR r s . guard p

968979

969980-- | Relational domain.

970981--

971982-- For injective relations, 'domain' and 'ker'nel coincide,

972−-- since @'ker' r ``sse`` 'identity'@ in such situations.

983+-- since @'ker' r ``sse`` 'id'@ in such situations.

973984domain ::

974985 ( CountableN a,

975− FromListsN a a

986+ FLN a a

976987 ) => Relation a b -> Relation a a

977−domain r = ker r `intersection` identity

988+domain r = ker r `intersection` id

978989

979990-- | Relational range.

980991--

981992-- For functions, 'range' and 'img' (image) coincide,

982−-- since @'img' f ``sse`` 'identity'@ for any @f@.

993+-- since @'img' f ``sse`` id@ for any @f@.

983994range ::

984995 ( CountableN b,

985− FromListsN b b

996+ FLN b b

986997 ) => Relation a b -> Relation b b

987−range r = img r `intersection` identity

998+range r = img r `intersection` id

988999

9891000-- Relation pretty print

9901001

9911002-- | Relation pretty printing

992−pretty :: (CountableDimensionsN a b) => Relation a b -> String

1003+pretty :: (CountableDimsN a b) => Relation a b -> String

9931004pretty (R a) = I.pretty a

9941005

9951006-- | Relation pretty printing

996−prettyPrint :: (CountableDimensionsN a b) => Relation a b -> IO ()

1007+prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()

9971008prettyPrint (R a) = I.prettyPrint a

src/LAoP/Utils.hs+4−1

@@ -96,7 +96,10 @@ module LAoP.Utils

9696 coerceNat3,

9797

9898 -- * List data type

99− List (..)

99+ List (..),

100+

101+ -- * Category type-class

102+ Category(..)

100103 )

101104where

102105

src/LAoP/Utils/Internal.hs+28−2

@@ -1,9 +1,9 @@

1+{-# LANGUAGE TypeFamilies #-}

12{-# LANGUAGE DeriveGeneric #-}

23{-# LANGUAGE TypeApplications #-}

34{-# LANGUAGE DataKinds #-}

45{-# LANGUAGE FlexibleInstances #-}

56{-# LANGUAGE GeneralizedNewtypeDeriving #-}

6−{-# LANGUAGE KindSignatures #-}

77{-# LANGUAGE ScopedTypeVariables #-}

99module LAoP.Utils.Internal

@@ -19,7 +19,10 @@ module LAoP.Utils.Internal

1919 coerceNat3,

2020

2121 -- * 'List' data type

22− List (..)

22+ List (..),

23+

24+ -- * Category type class

25+ Category(..)

2326 )

2427where

2528

@@ -30,6 +33,9 @@ import Data.Maybe

3033import GHC.TypeLits

3134import Control.DeepSeq

3235import GHC.Generics

36+import Data.Kind

37+import Prelude hiding (id, (.))

38+import qualified Prelude

3339

3440-- | Wrapper around 'Int's that have a restrictive semantic associated.

3541-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'

@@ -168,3 +174,23 @@ instance

168174 fromEnum (L x) =

169175 let as = [minBound .. maxBound]

170176 in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)

177+

178+-- | Constrained category instance

179+class Category k where

180+ type Object k o :: Constraint

181+ type Object k o = ()

182+ id :: Object k a => k a a

183+ (.) :: k b c -> k a b -> k a c

184+

185+{-# RULES

186+"identity/left" forall p .

187+ id . p = p

188+"identity/right" forall p .

189+ p . id = p

190+"association" forall p q r .

191+ (p . q) . r = p . (q . r)

192+ #-}

193+

194+instance Category (->) where

195+ id = Prelude.id

196+ (.) = (Prelude..)

stack-8.8.3.yamlnew file+1

@@ -0,0 +1 @@

1+resolver: lts-15.5

stack.yaml+1−66

@@ -1,66 +1 @@

1−# This file was automatically generated by 'stack init'

2−#

3−# Some commonly used options have been documented as comments in this file.

4−# For advanced use and comprehensive documentation of the format, please see:

5−# https://docs.haskellstack.org/en/stable/yaml_configuration/

6−

7−# Resolver to choose a 'specific' stackage snapshot or a compiler version.

8−# A snapshot resolver dictates the compiler version and the set of packages

9−# to be used for project dependencies. For example:

10−#

11−# resolver: lts-3.5

12−# resolver: nightly-2015-09-21

13−# resolver: ghc-7.10.2

14−#

15−# The location of a snapshot can be provided as a file or url. Stack assumes

16−# a snapshot provided as a file might change, whereas a url resource does not.

17−#

18−# resolver: ./custom-snapshot.yaml

19−# resolver: https://example.com/snapshots/2018-01-01.yaml

20−resolver: lts-14.18

21−

22−# User packages to be built.

23−# Various formats can be used as shown in the example below.

24−#

25−# packages:

26−# - some-directory

27−# - https://example.com/foo/bar/baz-0.0.2.tar.gz

28−# subdirs:

29−# - auto-update

30−# - wai

31−packages:

32−- .

33−# Dependency packages to be pulled from upstream that are not in the resolver.

34−# These entries can reference officially published versions as well as

35−# forks / in-progress versions pinned to a git hash. For example:

36−#

37−# extra-deps:

38−# - acme-missiles-0.3

39−# - git: https://github.com/commercialhaskell/stack.git

40−# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a

41−#

42−# extra-deps: []

43−

44−# Override default flag values for local packages and extra-deps

45−# flags: {}

46−

47−# Extra package databases containing global packages

48−# extra-package-dbs: []

49−

50−# Control whether we use the GHC we find on the path

51−# system-ghc: true

52−#

53−# Require a specific version of stack, using version ranges

54−# require-stack-version: -any # Default

55−# require-stack-version: ">=2.1"

56−#

57−# Override the architecture used by stack, especially useful on Windows

58−# arch: i386

59−# arch: x86_64

60−#

61−# Extra directories used by stack for building

62−# extra-include-dirs: [/path/to/dir]

63−# extra-lib-dirs: [/path/to/dir]

64−#

65−# Allow a newer minor version of GHC than the snapshot specifies

66−# compiler-check: newer-minor

1+resolver: lts-14.3

stack.yaml.lock+4−4

@@ -6,7 +6,7 @@

66packages: []

77snapshots:

88- completed:

9− size: 524789

10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml

11− sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2

12− original: lts-14.18

9+ size: 523878

10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml

11+ sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c

12+ original: lts-14.3

test/Examples/Quantum.hsnew file+31

@@ -0,0 +1,31 @@

1+{-# LANGUAGE StandaloneDeriving #-}

2+module Examples.Quantum where

4+import LAoP.Utils

5+import LAoP.Matrix.Type

6+import Data.Complex

7+import Prelude hiding (id, (.))

9+deriving instance Ord a => Ord (Complex a)

10+

11+xor :: (Bool, Bool) -> Bool

12+xor (False, b) = b

13+xor (True, b) = not b

14+

15+ker :: Num e => Matrix e a b -> Matrix e a a

16+ker m = tr m . m

17+

18+cnot :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

19+cnot = kr fstM (fromF xor)

20+

21+ccnot :: (Num e, Ord e) => Matrix e ((Bool, Bool), Bool) ((Bool, Bool), Bool)

22+ccnot = kr fstM (fromF f)

23+ where

24+ f = xor . tp (uncurry (&&)) id

25+ tp f g (a,b) = (f a, g b)

26+

27+had :: Matrix (Complex Double) Bool Bool

28+had = (1/sqrt 2) .| fromLists [[1, 1], [1, -1]]

29+

30+bell :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

31+bell = cnot . (had >< iden)

app/Examples.hs → test/Examples/Readme.hsrenamed+56−31

@@ -1,10 +1,10 @@

1+{-# LANGUAGE GADTs #-}

12{-# LANGUAGE DeriveGeneric #-}

23{-# LANGUAGE FlexibleContexts #-}

3−{-# LANGUAGE TypeOperators #-}

44{-# LANGUAGE TypeApplications #-}

55{-# LANGUAGE DataKinds #-}

7−module Examples

7+module Examples.Readme

88 ( exec

99 )

1010 where

@@ -15,13 +15,12 @@ import LAoP.Utils

1515import LAoP.Dist

1616import GHC.TypeLits

1717import Data.Coerce

18−import GHC.Generics

19−import Control.Category hiding (id)

20−import Prelude hiding ((.))

18+import qualified GHC.Generics as G

19+import Prelude hiding (id, (.))

2120

2221-- Monty Hall Problem

2322data Outcome = Win | Lose

24− deriving (Bounded, Enum, Eq, Show, Generic)

23+ deriving (Bounded, Enum, Eq, Show, G.Generic)

2524

2625switch :: Outcome -> Outcome

2726switch Win = Lose

@@ -31,7 +30,7 @@ firstChoice :: Matrix Double () Outcome

3130firstChoice = col [1/3, 2/3]

3231

3332secondChoice :: Matrix Double Outcome Outcome

34−secondChoice = fromF' switch

33+secondChoice = fromF switch

3534

3635-- Dice sum

3736

@@ -40,7 +39,7 @@ type SS = Natural 1 6 -- Sample Space

4039sumSS :: SS -> SS -> Natural 2 12

4140sumSS = coerceNat (+)

4241

43−sumSSM = fromF' (uncurry sumSS)

42+sumSSM = fromF (uncurry sumSS)

4443

4544condition :: (Int, Int) -> Int -> Int

4645condition (fst, snd) thrd = if fst == snd

@@ -50,38 +49,64 @@ condition (fst, snd) thrd = if fst == snd

5049conditionSS :: (SS, SS) -> SS -> Natural 3 18

5150conditionSS = coerceNat2 condition

5251

53−conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die

52+conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die

5453

5554die :: Matrix Double () SS

5655die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]

5756

5857-- Sprinkler

58+data G = Dry | Wet

59+ deriving (Bounded, Enum, Eq, Show, G.Generic)

5960

60−rain :: Matrix Double () Bool

61−rain = col [0.8, 0.2]

61+data S = Off | On

62+ deriving (Bounded, Enum, Eq, Show, G.Generic)

6263

63−sprinkler :: Matrix Double Bool Bool

64−sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]

64+data R = No | Yes

65+ deriving (Bounded, Enum, Eq, Show, G.Generic)

6566

66−grass :: Matrix Double (Bool, Bool) Bool

67−grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]

67+rain :: Matrix Double () R

68+rain = matrixBuilder gen

69+ where

70+ gen (_, No) = 0.8

71+ gen (_, Yes) = 0.2

72+

73+sprinkler :: Matrix Double R S

74+sprinkler = matrixBuilder gen

75+ where

76+ gen (No, Off) = 0.6

77+ gen (No, On) = 0.4

78+ gen (Yes, Off) = 0.99

79+ gen (Yes, On) = 0.01

80+

81+grass :: Matrix Double (S, R) G

82+grass = matrixBuilder gen

83+ where

84+ gen ((Off, No), Dry) = 1

85+ gen ((Off, Yes), Dry) = 0.2

86+ gen ((On, No), Dry) = 0.1

87+ gen ((On, Yes), Dry) = 0.01

88+ gen ((Off, No), Wet) = 0

89+ gen ((Off, Yes), Wet) = 0.8

90+ gen ((On, No), Wet) = 0.9

91+ gen ((On, Yes), Wet) = 0.99

92+

93+tag f = kr f id

6894

69−state :: Matrix Double () (Bool, (Bool, Bool))

70−state = khatri grass identity . khatri sprinkler identity . rain

95+state g s r = tag g . tag s . r

7196

72−grass_wet :: Matrix Double (Bool, (Bool, Bool)) One

73−grass_wet = row [0,1] . kp1

97+grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One

98+grass_wet state = row [0,1] . fstM . state

7499

75−rainning :: Matrix Double (Bool, (Bool, Bool)) One

76−rainning = row [0,1] . kp2 . kp2

100+rainning :: Matrix Double (G, (S, R)) One

101+rainning = row [0,1] . sndM . sndM

77102

78103-- Alcuin Puzzle

79104

80105data Being = Farmer | Fox | Goose | Beans

81− deriving (Bounded, Enum, Eq, Show, Generic)

106+ deriving (Bounded, Enum, Eq, Show, G.Generic)

82107

83108data Bank = LeftB | RightB

84− deriving (Bounded, Enum, Eq, Show, Generic)

109+ deriving (Bounded, Enum, Eq, Show, G.Generic)

85110

86111eats :: Being -> Being -> Bool

87112eats Fox Goose = True

@@ -96,21 +121,21 @@ cross LeftB = RightB

96121cross RightB = LeftB

97122

98123crossR :: R.Relation Bank Bank

99−crossR = R.fromF' cross

124+crossR = R.fromF cross

100125

101126-- | Initial state, everyone in the left bank

102127locationLeft :: Being -> Bank

103128locationLeft _ = LeftB

104129

105130locationLeftR :: R.Relation Being Bank

106−locationLeftR = R.fromF' locationLeft

131+locationLeftR = R.fromF locationLeft

107132

108133-- | Initial state, everyone in the right bank

109134locationRight :: Being -> Bank

110135locationRight _ = RightB

111136

112137locationRightR :: R.Relation Being Bank

113−locationRightR = R.fromF' locationRight

138+locationRightR = R.fromF locationRight

114139

115140-- Properties

116141

@@ -127,7 +152,7 @@ inv :: R.Relation Being Bank -> Bool

127152inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)

128153 where

129154 farmer :: R.Relation Being Being

130− farmer = R.fromF' (const Farmer)

155+ farmer = R.fromF (const Farmer)

131156

132157-- Arbitrary state

133158bankState :: Being -> Bank -> Bool

@@ -146,14 +171,14 @@ exec = do

146171 putStrLn "Monty Hall Problem solution:"

147172 prettyPrint (secondChoice . firstChoice)

148173 putStrLn "\n Sum of dices probability:"

149− prettyPrint (sumSSM `comp` khatri die die)

174+ prettyPrint (sumSSM . kr die die)

150175 putStrLn "\n Conditional dice throw:"

151176 prettyPrint conditionalThrows

152177 putStrLn "\n Checking that the last result is indeed a distribution: "

153− prettyPrint (bang . sumSSM . khatri die die)

178+ prettyPrint (bang . sumSSM . kr die die)

154179 putStrLn "\n Probability of grass being wet:"

155− prettyPrint (grass_wet . state)

180+ prettyPrint (grass_wet (state grass sprinkler rain))

156181 putStrLn "\n Probability of rain:"

157− prettyPrint (rainning . state)

182+ prettyPrint (rainning . state grass sprinkler rain)

158183 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"

159184 print (inv bankStateR)

test/Examples/Spreadsheets.hsnew file+47

@@ -0,0 +1,47 @@

1+{-# LANGUAGE DeriveGeneric #-}

3+module Examples.Spreadsheets

5+import LAoP.Utils

6+import LAoP.Matrix.Type

7+import GHC.Generics

8+import Prelude hiding (id, (.))

10+data Student = Student1 | Student2 | Student3 | Student4

11+ deriving (Eq, Show, Enum, Bounded, Generic)

12+

13+data Question = Question1 | Question2 | Question3 | Question4

14+ deriving (Eq, Show, Enum, Bounded, Generic)

15+

16+data Results = Exam | Test | Final

17+ deriving (Eq, Show, Enum, Bounded, Generic)

18+

19+test :: Matrix Float One Results

20+test = point Test

21+

22+exam :: Matrix Float One Results

23+exam = point Exam

24+

25+final :: Matrix Float One Results

26+final = point Final

27+

28+m :: Matrix Float Question Student

29+m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]

30+

31+w :: Matrix Float Question One

32+w = fromLists [[0.2, 0.3, 0.2, 0.3]]

33+

34+xls :: Matrix Float Question One

35+ -> Matrix Float Question Student

36+ -> Matrix Float Student One

37+ -> Matrix Float (Either Question Results) (Either One Student)

38+xls w m t = join (fork w m) (fork zeros r)

39+ where

40+ rExam = m . tr w

41+ rTest = tr t

42+ rFinal = rTest `maxPW` rExam

43+ r = (rExam . tr exam) + (rTest . tr test) + (rFinal . tr final)

44+

45+-- | Overloaded, point-wise 'max' function

46+maxPW :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b

47+maxPW = zipWithM max

test/Spec.hs+7−1

@@ -1,2 +1,8 @@

1+module Main (main) where

3+import qualified Examples.Readme as RD

15main :: IO ()

2−main = putStrLn "Test suite not yet implemented"

6+main = do

7+ putStrLn ("Testing if README works" :: String)

8+ RD.exec